No.
Not necessarily. Men kill women daily, and their actions are frequently shrugged off, explained away or are excused.
To my mind, gender would only arise (as a motivation for conviction) in such a situation if Chauvin himself was black, if he was was African-American, or, if the lady was both white and - and the "and" is of considerable importance here - impeccably middle class.
To me, the more interesting question arises when we take a look at that Venn diagram intersection of ultimate privilege: White, male, and middle class.
Now, if Chauvin had killed someone who was 1) white, 2) male, and 3) middle class, then, yes, we might see some expression of Republican outrage.