WASHINGTON—New Mexico Sen. Tom Udall announced Monday he wouldn’t seek re-election for a third term, the first Democrat to announce his retirement in the 2020 cycle.

“The worst thing anyone in public office can do is believe the office belongs to them, rather than to the people they represent,” Mr. Udall said in a statement. “That’s why I’m announcing today that I won’t be seeking re-election next year.”

Mr....