Same-sex or no, marrying under the tower will be difficult. Most of the answers focused on the practicality of marrying in Paris as a foreigner.
While that may be possible (especially as EU citizens), the ceremony would take place in some government office - the official won't come to the Eiffel tower with you.
However what you can do is to have the legal marriage in your home country or some place where it is convenient for you.
You could then come to Paris and have some form of ceremony under or near the Eiffel tower. I even found a website that offers "vow ceremonies" close to the Eiffel tower as a service - obviously those are not legal marriages, but are as close as you get.
You could also just bring someone you know (minister, friend, relative) to "officiate" the vows and roll your own ceremony.
Note that it has been pointed out in the comments that any religious marriage rites need to be performed after the civil union and any French priest/minister/imam/rabbi will supposedly check your marriage records.
A generic "vow" ceremony without any official religious and legal meaning - like the ones you find on the web - will probably fine regardless.
Note that even those "vow ceremonies" seem to take place somewhere near the Eiffel tower - the area under the tower is now cordoned off as an anti-terrorist measure and only accessible with a ticket; the place will usually be crowded with tourists and may not be the best spot to hold a ceremony with guests.
You can still have the ceremony close to the tower and then go for a visit, though. Or, if it's just the two of you you can just go and say the vows to each other.