American marrying British fiance in Gibraltar on UK tourist visa

I did not get a Marriage Visitor Visa prior to this trip because we had not been planning on getting married, but we are considering it now that I am in the UK.

You can get married in the UK on a standard visitor visa. What you cannot do, and what you need a Marriage Visitor Visa for, is to enter the UK with the intention of getting married there. But you didn't do that.

Thanks to Traveller for posting a relevant link in a comment. From that page:

If you do not have a marriage visitor visa or family visa

You can still give notice of your intention to get married or form a civil partnership but the immigration authorities at the Home Office will be told.

The Home Office might:

• ask questions about you and your relationship - if this happens you may need to wait up to 70 days before getting married or forming a civil partnership
• decide not to approve your notice - if this happens you cannot get married or form a civil partnership in the UK

The point of this is to identify and prevent immigration fraud. If you can show that you don't intend to settle in the UK in the immediate future, the Home Office should not delay or prevent your wedding.

Furthermore, as explained in a page helpfully linked by canonacer, if you still want to get married in Gibraltar, it seems as though you can do that:

There are no residency or nationality requirements.

I note, however, that there are two problems with that page:

1. As noted above, a fiancee visa would not have been necessary to marry in the UK, because the couple were not planning to settle in the UK. A Marriage Visitor Visa would have sufficed (and, as explained, is not necessary in your case).

2. It is similarly not fraudulent for a non-US national to marry a US citizen in the US while in visitor status when the couple plan to leave the country before the expiration of the non-US spouse's status.

In both of the above cases, though, some applicants might have difficulty proving their intent to leave the country after getting married, which could lead to refusal of a UK Marriage Visitor Visa or, for the US, refusal of entry as a visitor. That seems unlikely in the case described in the article, but it is something that couples should consider, especially if they haven't yet put down firm roots in a foreign place of residence.