I will be in Paris on Sunday, September 17th. I'm a huge Philadelphia
Eagles fan and if possible, the Mrs. and I want to watch their game
against the Giants. Are there any places in Paris that show NFL
football?

The match is shown by NASN (= North American Sports Network)
http://www.nasn.com/content/nasn/portal.nsf/systemcontent/aboutnasneu

i.e. on Sun 17 Sep 19:00 (local time)
NY Giants vs Philadelphia - NFL
http://www.nasn.com/content/nasn/portal.nsf/systemcontent/tvschedule

"NASN-Europe is available through various digital cable operators"
http://www.nasn.com/content/nasn/portal.nsf/systemcontent/faq#How_do_I_receive_NASN?

Apparently, you have to send an e-mail to find out more about the
cable operators in question:
http://www.nasn.com/content/nasn/portal.nsf/systemcontent/franceresponse

But, I suppose, there should be other ways of finding out where
exactly in Paris the match is being shown.

Contrary to my previous comment I now have the impression that the
match will not be shown in France by NASN, i.e. according to the
discussion in this forum:
http://www.universfreebox.com/article1597.html

I'll try to find out more.

Would this be an acceptable alternative?

http://sports.yahoo.com/nflgamepass

Prepare to be disappointed wherever you go.  My Giants are going to
crush your Eagles.

I would recommend to check the Long Hop, Métro Maubert-Mutualité for NFL.
But the best is to go directly to http://www.comptoirdessports.com
It is the program of sport events shown by bars in Paris. Although NFL
is not the most present sport in the list, you will find it most of
the time.

(And you will also get more info on the Long-Hop on this website.)

Fabrice