No. Johannes Kepler published what is now known as his third law of planetary motion in 1619 (in his treatise Harmonices Mundi), but discovered it already on May 15, 1618. He simply related mean distance of a planet from the Sun to its mean angular motion, without a word about a mass, I think. He did write on gravity and mass (not the precise physical term) in a foreword to his earlier book Astronomia Nova.
Thanks to people (Rafael Gil Brand, Roger Ceragioli and R. H. van Gent) from H-ASTRO discussion forum I have the following update #1:
1, The original form of the third law (formulated for planets), freely traslated to English reads approximately:
"...it is absolutely certain and perfectly correct, that the ratio which exists between the periodic times of any two planets is precisely 3/2 of the ratio of the mean distances, i.e. of the spheres themselves, bearing in mind, however, that the arithmetic mean between both diameters of the elliptic orbit is slightly less than the longer diameter"
2, Although (as far as I know from my own experience with early observations of double stars by Galileo) it is virtually impossible to prove that an earlier observation/idea didn't exist, it seems that the first application of Kepler's Third Law to the Jovian satellite system, is found in Newton's Philosophiae Naturalis Principia Mathematica (2nd ed. of 1713), lib. III, prop. 8, resulting in 1/1033 solar mass.
It is possible that Riccioli had something about the topic in one of his monumental treatises published around the middle of the 17th century.
Update #2
Riccioli seem to discuss relation between elongation of Galilean satellites of Jupiter and their orbital periods both in his Almagestum Novum and Astronomia Reformata, and cites Vendelinus (Godefroy_Wendelin). The Wikipedia entry for him states:
"In 1643 he recognized that Kepler's third law applied to the satellites of Jupiter."
without further details.
Update #3 - Final answer
I repost here the final answer by Christopher Linton from H-ASTRO:
"Kepler, in the Epitome of Copernican Astronomy (1618-1621), did apply his third law to the Jovian satellites (in Art. 553). He got the data from Simon Mayr's World of Jupiter (Mundus Jovialis, 1614). He establishes that $T^2/a^3$ is roughly constant and concludes that the physical mechanism which causes the planets to move as they do is the same as that which causes the Jovian satellites to rotate around Jupiter."
In general, yes you need to know the orbital inclination angle $i$ in order to fully solve the orbit. The radial velocity amplitude $K$ is just modified to $K \sin i$ (where $i=0$ is a face-on orbit). Combining this with the orbital period and Keplerian orbits gives you the "mass function"
$$ \frac{M_1^3 \sin^3 i}{\left(M_1 + M_2\right)^2} = \frac{K_{2}^3 \sin^3 i\ P_{orb}}{2\pi G},$$
where the right hand side can be measured from radial velocity data in a spectroscopic binary. If you have a velocity amplitude for both stars, then there is a similar expression with the labels reversed. Without $i$ this can then only tell you the mass ratio $M_1/M_2$.
There are several ways to break this degeneracy depending on what kind of binary system it is.
In a visual binary system where you can observe the orbits, then the orbital path of both objects can be observed and the inclination of the orbit is directly measured. However, radial velocity amplitudes are not usually measurable (too small) and one relies on the absolute size of the orbit, which in turn requires a distance (parallax) estimate.
In an eclipsing binary, then the shape and depth of the eclipses can be uniquely solved to give the inclination and hence the masses of the individual stars.
In non-eclipsing close binary systems, or when one component is not seen, then ellipsoidal modulation of the seen component depends on the mass ratio and the inclination. Together with the radial velocity curve, this can then give unique masses for the components.
In general it is not possible to get any more than a mass ratio for the components of a double lined spectroscopic binary system (SB2), or the "mass function" (see above) of a single lined spectroscopic binary system (SB1).
To make further progress in these general cases you need an estimate of the primary mass. This can be done with reference to stellar evolutionary models. In principle, for an SB2, the mass ratio and the combined appearance of an object in the Hertzsprung-Russell diagram contain enough information to determine the masses of the individual components and the age of the system. In practice this is hard and there are degeneracies. A better way is to fit a combination of spectral type templates to the measured spectrum and hence estimate the spectral types and hence masses.
In an SB1 you really are stuck. The spectral type and position in the HR diagram give you $M_1$, but you will only have a lower limit to the unseen secondary mass. This is why it is difficult to estimate the masses of black holes in binaries - you need to know the inclination.
Best Answer
You can use the center of mass formula.
Set the origin of your coordinate system at the center of the Earth, then $\vec{r}_1 = \vec{0}$ and $\vec{r}_2 = d$ and $$r_{center} = \frac{m_1r_1+m_2r_2}{m_1+m_2} = \frac{m_2}{m_1+m_2} \cdot d$$ as you have as well.