Predictably, politicians often play this song at campaign stops to indicate their resilience. When George W. Bush used it during his 2000 presidential campaign, Petty threatened to sue, as he found Bush odious. Bush stopped using the song (guess he does back down) but won the election, which was decided when Petty's home state of Florida went to Bush by a very slim margin. The night Bush's opponent, Al Gore, conceded the election (December 13, 2000), Petty played the song for him at Gore's home
After President Donald Trump used "I Won't Back Down" at a highly publicized campaign rally in Tulsa, Oklahoma, on June 20, 2020, Petty's widow Dana, ex-wife Jane, and daughters Adria and Annakim issued a statement making it clear they did not approve. "Both the late Tom Petty and his family firmly stand against racism and discrimination of any kind," they wrote. "Tom Petty would never want a song of his used for a campaign of hate. He liked to bring people together. Tom wrote this song for the underdog, for the common man and for EVERYONE."
Heartbreakers keyboard player Benmont Tench
agreed, posting, "I in no way approve of Trump even whistling any piece of music associated with our band."
Petty's family joined R.E.M., Elton John and several others in opposing Trump's use of their songs, but legally, there is nothing they can do to stop him as long as the venues have the proper licensing. These statements are more about going on record to oppose a politician.