No, Hawkeye did not kill Black Widow.

Why did Barton kill Natasha?

Background. S.H.I.E.L.D. had managed to track down Natasha Romanoff, a dangerous assassin and the Red Room operative known as Black Widow, and tasked Hawkeye to find and eliminate her. ... S.H.I.E.L.D. decided that Dreykov's influence made him too dangerous, so they ordered Barton to take him out.

How did Hawkeye kill Natasha?

Unrepentant and knowing her former ally has no intention of sparing her life, Natasha simply asks Clint if he has a message she can pass along to his wife and children. In response, Hawkeye tells her she won't be seeing them where she is going - and fires an arrow into her head.