"Medicare for all is a goal that we should all be fighting for," Gillibrand said.

Gillibrand's call was held moments after Senate Republican leaders withdrew the bill sponsored by U.S. Sens. Lindsey Graham and Bill Cassidy to repeal significant portions of the Affordable Care Act, a 2010 health care law also referred to as "Obamacare."

The Graham-Cassidy bill didn't have enough support among GOP senators to pass. Over the last few days, a handful of Republicans came out against the proposal, including U.S. Sens. Susan Collins and John McCain — both of whom opposed the last GOP effort.

Gillibrand credited opponents who "made clear that they would be harmed if that bill passed" for helping derail the Republicans' health care repeal measure.

"While they aren't moving forward with this particular version right now, we know this isn't the end of the fight," she said. "They're going to keep bringing this awful legislation up one way or another. It's the wrong approach."

Following the failure of the GOP health care bill, Gillibrand is hopeful that Congress will focus on bipartisan solutions. She is one of several Democrats who have acknowledged that the Affordable Care Act is flawed and the cost of health insurance must be addressed.