I think you can get married in the states if you wanted to (even if that's sort of defeating the purpose of the fiance visa...but people change their mind!), but as you're already here on a fiance visa, why not just get married at a register office in the UK? It'll be much cheaper than flying back to the states, that's for sure.
If you get married in the States, you'd have to apply for a spousal visa before re-entering the UK. If you get married in the UK, you just need to fill out form FLR(M) and send that in or make an in-person appointment.