Unfortunately, you answered your own question. Removing the video (or at least the audio track of it) is the easiest answer. The use of someone's recorded song does amount to copyright infringement. The copyright holder (normally a record label) would be allowed to file suit against you. While that is unlikely, Facebook could stop you from having a page or from posting any further. It is a private company, not a public space
I agree with Oscar. Your use of the copyrighted song is an infringement of not only the sound recording copyright, as Oscar points out, but the copyright in the underlying musical composition. You are, in fact, doing something "wrong" in that you are using another person's intellectual property without their permission. In the physical realm, it would be similar to your neighbor helping himself to your lawnmower without your permission. Here, you're using intellectual property that belongs to someone else. I also wanted to point out that Facebook is probably contacting you because it has likely been contacted by the record label and/or music publisher asking that the infringing video be removed. This is how copyright holders enforce their rights - they contact the ISP or website developer via a cease and desist letter.
My advise is to simply replace the audio in the video with something that is freely available. I have provided a great blog post link below that lists places on the web where you can download "free" music, i.e., music to which the owner's give free license for you to use. Find something you like among the free stuff!