WASHINGTON—House Majority Leader Eric Cantor on Wednesday suggested that Republicans will continue a push to overhaul programs such as Medicare, saying in an interview that "promises have been made that frankly are not going to be kept for many" and that younger Americans will have to adjust.

"What we have to be I think focused on is truth in budgeting here," Mr. Cantor (R., Va.) told The Wall Street Journal's Opinion Journal. He said "the better way" for Americans is to "get the fiscal house in order" and "come to grips...