Mihawk would probably go back to farming with baboons and with Perona living in Kuraigana hopefully, maybe Buggy & his bunch too.
His character is of a loner who's too powerful and reached the top & then found emptyness in his life, much like Saitama from One Punch Man.
No wonder he doesnt understand why his past rival gave his arm (& potential) up for a kid.
The first time he finds that in his life is when Perona and Zoro live at Kuraigana.
To come full circle, he'd simply go back to finding happyness in his life after giving up his throne to a worthy challenger in Zoro.