I don't think SOPA has much to do with Internet governance. It doesn't contemplate any changes to IP addressing, DNS, or BGP. It doesn't really attempt to reach outside the US. (ACTA is another matter.)
shouldn't the faculties in charge of running it be global institutions as well?
This has been discussed in great length elsewhere, but I'll give a few points. While in principle of course the whole world should have a say in Internet governance, in reality all the specific proposals that I've seen are about making the Internet worse in some way. The good thing about ICANN is that it's so ineffectual and slow-moving that it can't cause much damage.
Consider the global telephone network which is regulated by the ITU, an organization that appears to be 100% captured by telcos, many of whom are national PTTs that make AT&T look friendly. Right now the ITU is trying to figure out how to take control of the Internet from ICANN so they can "recognize local values" (i.e. censorship) and make everything much more expensive by e.g. eliminating settlement-free peering.