Given a list of overtones (F1, F2, F3, etc), how do I compute the fundamental frequency? Can I do something like F2/F1=F1/F0? Is it the correct method to use?
The frequencies of the harmonics are integer multiples of the fundamental frequency $f_0$, i.e. $f_n = (n+1)f_0$. The fundamental frequency $f_0$ is the greatest common divisor of the harmonics $f_n$. If you are sure that there is no other unknown harmonic between two known harmonics, e.g. you know that you have the fourth and the fifth harmonic, then $f_0$ is of course the difference between the two. But if you just have a collection of harmonics and you don't know anything else about them, then you need to determine $f_0$ as the gcd of $f_n$.
Nope. Difference between overtones is a good point to start, i,e F3-F2, F2-F1. The differences should be all the same or multiple of each other. The smallest one is often the fundamental. It gets more tricky of the spectrum is "sparse", i.e. a lot of the harmonics are missing. Then you need to find a largest possible divisor that turns all frequencies into integers or, to be precise, so that the ratio of frequency to fundamental is within the measurement accuracy of the nearest integer.
Look up the Harmonic Product Spectrum algorithm, which given a sufficient number of actual overtones, is a bit more robust against missing overtones and added noise spectra, than just subtracting all successive tones frequency pairs.