He won't. Mihawk will be losing to Imu and then Zoro will save him and then defeat Imu, becoming the strongest swordsman. Then he eats a senzu bean, Tashigi appears and uses her new ability Deva, which counters Asura. But it doesn't matter, cuz she's still too weak, then she cries and her glasses get dirty. Zoro tells her to get lost, Tashigi says "no, u", Zoro says "I already am".
Zoro reveals he's originally from Raftel. He got lost on the way to preschool and ended up in East Blue, so he joined the future pirate king's crew to get back home.
Genuinely, he might not fight him. It seems hard to imagine they can set up Zoro vs Mihawk, Zoro vs Shiryu and Zoro vs some celestial dragon boss before the series ends. Also maybe Zoro vs Fujitora (unless Shiryu defeats Fujitora for hype).