WASHINGTON—House Republicans released their plan to cut Medicaid spending, with the program’s defenders in the GOP appearing to win the intraparty clash over how aggressively to change the system that provides health insurance to more than 70 million low-income and disabled people.

Republicans are steering clear of dramatic Medicaid cuts in their spending bill, instead opting for smaller changes backed by moderate and swing-state members who had balked at the prospect of slashing a program that covers more than 70 million low-income and disabled Americans.