Days after the Supreme Court delivered a victory for his health care law for the second time, President Obama flew into mostly Republican territory on Wednesday and began an aggressive push to get states that have resisted parts of the law to expand care to more of their poor residents.

“With the Supreme Court case now behind us, I’m hoping what we can do is focus on how to make it even better,” Mr. Obama said to an audience here that included health care administrators and people who got health insurance under the Affordable Care Act.