Short answer is, no not on this visit.

Longer answer is that you cannot apply for a fiancée or spouse visa while already in the country. Use the search tool to find other posts on this forum as well as look at the http://www.britainusa.com/ site for info on how to apply when you're back in the US.

Now from personal experience and following the experiences of others (including people coming in from the US) a lot of people find the UK makes them jump through a ludicrous number of hoops and find it simpler to marry in the foreign country and then to apply for a spouse visa once married.