TOPEKA, Kan. (AP) — Kansas' conservative governor vetoed a bill Thursday that would have extended the state's health coverage for the poor under former President Barack Obama's signature health care law, setting up a likely showdown with the Legislature.

Republican Gov. Sam Brownback argued that the proposed expansion of the state's Medicaid program would fail to serve "the truly vulnerable" ahead of "the able-bodied," and that it would burden the state budget with "unrestrainable entitlement costs."

When the Republican-controlled Legislature gave final approval to the measure Tuesday, it had...