1st comment:
It's worth thinking for a second about where Wick rotation comes from. You can do this in the context of the quantum mechanics of a free particle. In QFT, all of the details are more complicated, but the basic idea is the same.
In free particle, QM, we get the path integral by inserting sums over intermediate states at various times. The need for Wick rotation arises as soon as you do this just once.
$\langle q' | e^{- \frac{ iP^2 t}{2m\hbar}}|q\rangle = \int_{-\infty}^\infty \langle q'| p \rangle \langle p |e^{- \frac{ iP^2 t}{2m\hbar}}|q\rangle dp = \frac{1}{2\pi\hbar} \int_{-\infty}^\infty e^{\frac{-i t }{2m\hbar} p^2 + i\frac{q' - q}{\hbar} p} dp$.
This is an oscillatory integral. The integrand has norm 1 because the argument of the exponential is purely imaginary. Such integrals don't converge absolutely, so the right hand side of this equation is not obviously well-defined. It's not Lebesgue integralable, although it is convergent as a Riemann integral, thanks to some rather delicate cancellations. To make the integral well defined -- equivalently to see how these cancellations occur -- we need to supply some additional information.
Wick rotation provides a way of doing this. You observe that the left hand side is analytic in $t$, and that the right hand side is well-defined if $Im(t) < 0$. Then you can define the integral for real $t$ by saying that it's analytic continued from complex $t$ with negative imaginary part.
2nd comment:
As V. Moretti pointed out, in QFT, it's in some sense backwards to think of analytically continuing from Minkowski signature to Euclidean signature. Rather, one finds something in Euclidean signature which has nice properties and then analytically continues from Euclidean to Minkowski. However, one can often begin this process by taking a Minkowski action and finding its Euclidean version, and then trying to build up a QFT from there. There's no guarantee that this will work though. Spinor fields may have reality conditions that depend on the signature of spacetime. Or the Euclidean action you derive may be badly behaved. This is famously the case for Einstein's gravity; the Euclidean action is not bounded below, so one does not get a sensible Euclidean theory.
TL;DR: OP's title question (v7) about instantons in the Minkowski signature is physically meaningless. It is an irrelevant mathematical detour run amok. The connection to physics/Nature is established via a Wick rotation of the full Euclidean path integral, not bits and pieces thereof. Within the Euclidean path integral, it is possible to consistently expand over Euclidean instantons, but it meaningless to Wick rotation the instanton picture to Minkowski signature.
In more details, let there be given a double-well potential
$$V(x)~=~\frac{1}{2}(x^2-a^2)^2. \tag{A}$$
The Minkowskian and Euclidean formulations are connected via a Wick rotation
$$ t^E e^{i\epsilon}~=~e^{i\frac{\pi}{2}} t^M e^{-i\epsilon}.\tag{B} $$
We have included Feynman's $i\epsilon$-prescription in order to help convergence and avoid branch cuts and singularities. See also this related Phys.SE post.
I) On one hand, the Euclidean partition function/path integral is
$$\begin{align} Z^E~=~&Z(\Delta t^E e^{i\epsilon})\cr
~=~& \langle x_f | \exp\left[-\frac{H \Delta t^E e^{i\epsilon}}{\hbar}\right] | x_i \rangle \cr
~=~&N \int [dx] \exp\left[-\frac{S^E[x]}{\hbar} \right],\end{align}\tag{C} $$
with Euclidean action
$$\begin{align} S^E[x]
~=~&\int_{t^E_i}^{t^E_f} \! dt^E \left[ \frac{e^{-i\epsilon}}{2} \left(\frac{dx}{dt^E}\right)^2+e^{i\epsilon}V(x)\right]\cr
~=~& \int_{t^E_i}^{t^E_f} \! dt^E \frac{e^{-i\epsilon}}{2} \left(\frac{dx}{dt^E}\mp e^{i\epsilon}\sqrt{2V(x)}\right)^2 \cr
&\pm \int_{x_i}^{x_f} \! dx ~\sqrt{2V(x)}.\end{align}\tag{D}$$
and real regular kink/anti-kink solution$^1$
$$\begin{align} \frac{dx}{dt^E}\mp e^{i\epsilon}\sqrt{2V(x)}~\approx~&0 \cr
~\Updownarrow~ & \cr
x(t^E)
~\approx~&\pm a\tanh(e^{i\epsilon}\Delta t^E). \end{align}\tag{E}$$
Note that a priori space $x$ and time $t^E$ are real coordinates in the path integral (C). To evaluate the Euclidean path integral (C) via the method of steepest descent, we need not complexify space nor time. We are already integrating in the direction of steepest descent!
II) On the other hand, the corresponding Minkowskian partition function/path integral is
$$\begin{align} Z^M~=~&Z(i \Delta t^M e^{-i\epsilon})\cr
~=~& \langle x_f | \exp\left[-\frac{iH \Delta t^M e^{-i\epsilon}}{\hbar} \right] | x_i \rangle\cr
~=~& N \int [dx] \exp\left[\frac{iS^M[x]}{\hbar} \right],\end{align}\tag{F} $$
with Minkowskian action
$$\begin{align} S^M[x]~=~&\int_{t^M_i}^{t^M_f} \! dt^M \left[ \frac{e^{i\epsilon}}{2} \left(\frac{dx}{dt^M}\right)^2-e^{-i\epsilon}V(x)\right]\cr
~=~& \int_{t^M_i}^{t^M_f} \! dt^M \frac{e^{-i\epsilon}}{2} \left(\frac{dx}{dt^M}\mp i e^{i\epsilon}\sqrt{2V(x)}\right)^2 \cr
&\pm i \int_{x_i}^{x_f} \! dx ~\sqrt{2V(x)},\end{align}\tag{G}$$
and imaginary singular kink/anti-kink solution
$$\begin{align} \frac{dx}{dt^M}\mp i e^{i\epsilon}\sqrt{2V(x)}~\approx~&0 \cr
~\Updownarrow~ & \cr
x(t^M)~\approx~&\pm i a\tan(e^{-i\epsilon}\Delta t^M)\cr
~=~&\pm a\tanh(i e^{-i\epsilon}\Delta t^M). \end{align}\tag{H}$$
It is reassuring that the $i\epsilon$ regularization ensures that the particle starts and ends at the potential minima:
$$ \lim_{\Delta t^M\to \pm^{\prime}\infty} x(t^M)~=~(\pm a) (\pm^{\prime} 1). \tag{I}$$
Unfortunately, that seems to be just about the only nice thing about the solution (H). Note that a priori space $x$ and time $t^M$ are real coordinates in the path integral (F). We cannot directly apply the method of steepest descent to evaluate the Minkowski path integral. We need to deform the integration contour and/or complexify time and space in a consistent way. This is governed by Picard-Lefschetz theory & the Lefschetz thimble. In particular, the role of the imaginary singular kink/anti-kink solution (H) looses its importance, because we cannot expand perturbatively around it in any meaningful way.
--
$^1$ The explicit (hyperbolic) tangent solution (E) is an over-simplified toy solution. It obscures the dependence of finite initial (and final) time $t^E_i$ (and $t^E_f$), moduli parameters, and multi-instantons. We refer to the literature for details.
Best Answer
The final condition for a realistic theory in physics is that its predictions (of scattering amplitudes or correlators etc.) have to agree with observations. This implies that the predictions have to be consistent and obey some general consistency conditions (unitarity, non-negativity of probabilities, some symmetries, locality or approximate locality, and so on).
For theoretical theories, it's just the general consistencies that hold. Now, the task is to classify all possible theories and learn how to calculate with them.
It turns out that the Euclidean spacetime or world sheet is simply a simpler, more straightforward, more free-of-subtleties approach to produce a machine that calculates some scattering amplitudes or other observables.
At least formally, the Euclidean theories may be continued to analytic ones and vice versa. For nontrivial spacetime topologies, the Euclidean objects are likely to be more manageable. For example, the world sheets in string theory (think about a torus or pants diagrams etc.) are much more well-behaved in the Euclidean signature so we may consider this approach "primary". Covariant calculations in string theory are almost always done with the Euclidean ones. The result may be continued to the Minkowski momenta etc. and some of the consistency conditions above are still guaranteed to hold because of some properties of the complex calculus.
In the light-cone gauge, we may work directly with the Minkowski-signature world sheets. But we pay the price that the interaction points where strings split or join are singular and the direction of the "future time" is ambiguous. We must also include contact terms, higher-order interaction terms, to deal with some divergences caused by the singular world sheets, but when these things are summed over, we may prove that the resulting amplitudes agree with the covariantly computed ones (in the Euclidean signature).
Gravity in $d\geq 4$ (and maybe 3) suffers from the "negative norm conformal factor". The Euclideanized Einstein-Hilbert action $\int R \sqrt{g}$ is no longer positively definite. In particular, if you consider scalar waves that scale the metric by an overall number, $g_{\mu\nu}=e^F\eta_{\mu\nu}$, and derive the kinetic term for $F$, it will have the opposite sign than the kinetic term for other components of the metric tensor (the physical polarizations of the gravitational waves, like $g_{xy}$).
It follows that the action will be bounded neither from below nor from above, and $\exp(-S_E)$ in the Euclidean path integral will diverge in some region of the configuration space. In this sense, people believe that the Minkowskian path integral must be the "more kosher one" for higher-dimensional gravity. But this is a bit empty statement because at the quantum level, higher-dimensional gravity obtained as a direct quantization of Einstein's equations is inconsistent, anyway. And string theory which is consistent and contains gravity doesn't give us any tool to directly rewrite the path integral in terms of spacetime fields including the metric; it is not a field theory in the ordinary sense. So the preference for the "Minkowski signature" is a bit vacuous. After all, the Minkowskian action isn't bounded from either side, either. This is considered "not to be a problem" because the integrand is $\exp(iS)$ which still has the absolute value equal to one, so it doesn't diverge. But I would personally say that the unboundedness of the Euclidean action is the "same" problem for the Minkowskian path integral.
Quite generally, the Wick rotation is extremely important in quantum field theory and it is actually even more important in quantum gravity or places with many spacetime (or world sheet) topologies, i.e. in situations where one has many different "time variables" in which we might try to expand things. One shouldn't be afraid but at the end, whatever theory he deals with, he gets some amplitudes whose self-consistency (and/or consistency with observations) must be verified. With some "good rules of behavior" while Wick-rotating, one may be "pretty sure" that some tests will be passed.