Take the example of a hydrogen atom.
The energy levels in the hydrogen atom are given by
$$ E_n = -2\pi^2 \frac{m_e^4}{n^2 h^2}.$$
The spacing between two energy levels determines the frequency of an emitted photon when a radiative transition is made between them:
$$ h \nu_{n_2\rightarrow n_1} = \frac{2\pi^2 m_e^4}{h^2}\left(\frac{1}{n_2^{2}} - \frac{1}{n_1^{2}}\right). $$
Thus the frequency of the transition will be proportional to $h^{-3}$. Thus if $h$ changes, then the frequency of spectral lines corresponding to atomic transitions would change considerably.
Now, when we look at distant galaxies we can identify spectral lines corresponding to the atomic transitions in hydrogen. As John Rennie says, these are redshifted and so this could be interpreted as a systematic spatial change in Planck's constant with distance from the Earth. However, this shift would have to be the same in all directions - since redshift appears to be very isotropic on large scales - and would thus place the Earth at the "centre of the universe", which historically has always turned out to be a very bad idea. Alternatively we could assume Planck's constant was the same everywhere is space but was changing with time - this would produce an isotropic signal. (NB: Cosmic redshift cannot be explained in this way since there are other phenomena, such as the time dilation of supernova light curves that do not depend on Planck's constant in the same way, that would be inconsistent).
That is not to say that these issues are not being considered. Most effort has been focused on seeing whether the fine structure constant $\alpha$ varies as we look back in time at distant galaxies. The reason for this is, as Count Iblis correctly (in my view) points out, the quest for a variable $h$ is futile, since any phenomenon we might try to measure is actually a function of the dimensionless $\alpha$, which is proportional to $e^2/hc$. We might claim that if either varied, it would produce a variation in $\alpha$, but this would amount to a choice of units and only the variation in $\alpha$ is fundamental.
So, in the example above, the hydrogen energy levels are actually proportional to $\alpha^2/n^2$. A change in $\alpha$ can be detected because the relativistic fine structure of spectral lines - i.e the separation in energy between lines in a doublet for instance, also depends on $\alpha^2$, but is different in atomic species with different atomic numbers.
There are continuing claims and counter claims (e.g. see Webb et al. 2011; Kraiselburd et al. 2013) that $\alpha$ variations of 1 part in a million or so may be present in high redshift quasars (corresponding to a fractional change of $\Delta \alpha/\alpha \sim 10^{-16}$ per year) but that the variation may have an angular dependence (i.e. a dependence on space as well as time).
Yes there is.
The solution to the Friedmann equation in a flat universe with a cosmological constant is
$$H^2 = \frac{8\pi G}{3}\rho + \frac{\Lambda}{3},$$
Thus, as the universe expands and the relative importance of gravitating matter, characterised by density $\rho$, decreases, then $\Lambda \simeq 3H^2$.
We are already (just) in a dark energy dominated universe, so the relationship is already (nearly) true.
Best Answer
$G$ is not exactly larger than $h$ by a factor of $10^{23}$ in SI units, as you are probably aware (just making sure). There is also no expected numerical relationship between the two that has a physical interpretation. You have to understand that these constants are mostly just due to our (to some extent) arbitrary choices of units. These are, of course, motivated by everyday convenience. But this doesn't mean that the commonly used SI units have any physical significance. In fact, there are several other unit systems. One particularly interesting one that is quite popular among physicists doing fundamental research is known as the Planck unit system.
In terms of Planck units, both $G$ and $\hbar$ are equal to $1$, as well as $c$, $k_B$ and $4\pi\epsilon_0$, the speed of light, Boltzmann's constant and the inverse of the Coulomb constant respectively. The Planck unit system attempts to eliminate the arbitrary choices due to the perspective of humans, which just so happen to live on certain energy, length, etc. scales. This is done by defining the units of measurement only in terms of fundamental constants of nature. The idea is that these constants are really what 'nature measures in', so setting their numerical value to $1$ makes sense. Related is the concept of a natural unit system, of which several exist. These all attempt to formulate things in a 'natural' sense (which, among other things, depends on the field of study).