If he used the exact music regardless of the words, he can be sued under music publishing copyright laws. He would also need to have what is called a sychronization license if the music is used in conjunction with a visual image such as a video. I was in the music business for many years and I had to deal with unauthorized use of copyrighted material often. Henley may be a jerk, but he has the legal right to be a jerk over words and/or music he has copyrighted.
You are wrong. You may have been in music, but I wrote TV comedy shows for many years and you cannot sue when your music, poetry, literature or movie content is used for satire or parody. It has nothing to do with publishing or copyright law. You can’t sue.