Eowyn destroyed the witch king.
Frodo destroyed Sauron and Gollum.
Gandalf destroyed the Balrog.
Aragorn did what any king would do
took the credit.
Could he best Eowyn, maybe, but he couldn't best the witch king, so maybe not. Of course he is old by the time he meets Eowyn, she could probably just out run him. Or feed him stew.