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.