Mihawk sought out the strong
swordsmen.
That's not why he hunted the marines. He holds a personal grudge against the marines. He hunted und killed them for the fun of it.
Stopped when he beat everyone daring to face him, no doubt that included admirals, and it got boring at some point. It literally needed an additional freelancer contract with privileges to make him stop humiliating the marines. At this point, most likely even the first shichibukai contract ever.
Let's stick to the manga and less to headcanon shall we?