Acquired by Kansas City from Cincinnati last July 26, the pitcher went 4-7 for the Royals during the regular season, then 2-1 in the postseason.
His two-hitter in Game 2 of the World Series against the New York Mets gave Kansas City a 2-1 series lead, and the Royals won in five games for their first championship since 1985.
Cueto left as a free agent after the season to sign a $130 million, six-year contract with the San Francisco Giants. He is 13-2 with a 2.53 ERA in 2016 and started for the National League in the All-Star Game.
"I believe that I earned it," he said of the White House trip, speaking through a translator after his start Saturday against the New York Yankees. "I was on the team. I did something for the team for them to win the World Series, so I was expecting it."
Cueto said he did not speak to anyone on the Royals about attending Thursday's ceremony at the White House.
"I feel OK. I feel normal, but I also feel that it's the team's, the organization's responsibility to invite me," he said. "I did a job for them, so I think that they should have had the courtesy to invite me."
Kansas City said it set limitations on which players would get invitations to the White House visit: players had to be on the 2015 team all year, on the active postseason roster and not on another team at this time.
Outfielder Jonny Gomes, another part-season Royals player in 2015, also expressed disappointment he wasn't asked to the White House.
"I've been sent down. I've been non-tendered, whatever," Gomes told Fox Sports last week. "But this is probably the worst news of my professional baseball career."
Information from The Associated Press was used in this report.