New York took the opener 7-4 behind Luis Severino's eight strong innings and a six-run fourth inning, then dropped the second game 4-2.
Judge struck out eight times in nine at-bats, the most in a doubleheader since strikeouts for hitters were first recorded in 1910 in the NL and 1913 in the AL, according to the Elias Sports Bureau. Judge struck out in each of his five at-bats in the late game, including with a runner aboard in the bottom of the ninth.
The five strikeouts were a career high for Judge, who led the majors with 208 strikeouts while winning AL Rookie of the Year last season.
"That's not a rough day, that's a terrible day," Judge said. "But the beauty of baseball is that I get to wake up to fresh, new at-bats tomorrow."
Information from The Associated Press was used in this report.