This is a common misconception but is not correct.

Anti-competitive behavior can be illegal all by itself. Price fixing is one obvious example but there are plenty of others.

In addition to most violations of antitrust law being from companies that do not have monopolies, it goes the other way too. You can have an actual monopoly without violating antitrust law.

If you got the monopoly without doing anti-competitive behavior, such as by simple having a better product or by getting lucky and having competitors that all made stupid mistakes that sunk them, your monopoly may be legal.

Correct Apple and several book publishers got in trouble for doing just that years ago in the fight against Amazon.

This is a patently false statement from first principles. Anticompetitive behavior to keep a monopoly is illegal, but that is not the only case. Cartels, predatory pricing, and price fixing are all illegal with or without monopolies and are all considered anticompetitive practices.