Conservatives don't have a corner on the Roman Catholic vote, an archbishop said while assessing the effect that Pope Francis will have on U.S. politics.

"[W]hat Pope Francis has done is reminded us of the latitude of Catholic beliefs and Catholic principles," Cardinal Timothy Dolan replied when NBC's David Gregory asked how the pope will influence "American politics, for instance, on some of these issues, be it abortion or gay marriage."

Dolan argued that it is wrong to "pigeonhole" Catholic bishops as conservative based on their social teachings regarding abortion, gay marriage and contraception.

"The church's teaching on that is unwavering," he said. "But that's not it."

The cardinal faulted President Obama for alienating Roman Catholics who would support Obamacare if not for certain policies, such as the contraception mandate.

"[T]hat's when we began to worry and draw back and say, 'Mr. President, please, you're really kind of pushing aside some of your greatest supporters here. We want to be with you, we want to be strong. And if you keep doing this, we're not going to be able to be one of your cheerleaders,' " Dolan said. "And that sadly is what happened."

He emphasized that passing an immigration bill opposed by many conservatives is just as much of a "moral imperative" as the pro-life policies supported by Roman Catholic Church teaching.

"On health care, we might be upset with the Democrats, with the administration," Dolan said. "On immigration, we're saying to the House of Representatives, which is dominated by the Republicans, 'You guys have got to get your act together.' And this is the best chance we've had in [years for] fair and just immigration reform. It's in your lap and it doesn't seem to be going anywhere. And we're not going to let you off the hook. So yeah, we're disappointed there as well."