Upvote:4
US immigration is weird.
You are technically allowed to fly to the US on the Visa Waiver Program, get married and fly out. (Visa Waiver Program or VWP is the correct name for what is usually called "coming on an ESTA".)
If you get married to a US citizen while in the US on VWP, then you are allowed to file for 'adjustment of status' to become a resident and to stay in the US while your application is processed.
However you are not allowed to come to the US on VWP with intent to remain by adjusting your status as a result of marriage.
This means your admission to the US depends entirely on your ability to convince US immigration that you will absolutely definitely leave the US after you are married. Given how many people see marriage to a US citizen as a path to residency (to the point where it is a staple of sitcoms) that may be very hard. US immigration is required to presume that you have immigrant intent unless you show them otherwise.
All the usual evidence of ties to your home country will help, starting with being a citizen of a rich European country, and also having a job, a house, a return ticket etc. I have no idea how much evidence will be required to do the convincing.