It depends on how you define martial art. Traditionally, martial arts are thought of as being combat systems of Chinese, Japanese or East Asian origin that were conceived for military training and warfare purposes. Under that definition, western boxing is not a martial art.
However, according to wikipedia, the term martial arts originated in European culture and is a reference to the Roman god Mars, who is the god of war. We know that boxing was one of the original sports practiced in the ancient Olympic games, and it was a fixture in Roman culture as well. And apparently, boxing training was at least somewhat incorporated in Greco-Roman military training, especially in Spartan culture.
So, according to that information, it would be considered a martial art and perhaps the oldest martial art in the world aside from Greco-Roman wrestling. Either way, it’s a combat system that requires intense dedication and decades to develop and master its skills. In this way, it’s no different than Judo, Karate, Muay Thai or any other martial art.
Still, I don’t think most boxers would consider it to be a martial art. There isn’t much spiritual or cultural aspects that are attached to boxing training that you see in other martial arts. Maybe someone like Manny Pacquaio or Naoya Inoue would disagree with that.