(CNN) Republican Sen. Rand Paul objected Wednesday to an attempt to pass the bill funding 9/11 first responders' health care unanimously, arguing that passing such a long-term bill without offsetting the cost would contribute to the national debt.

When presidential candidate and Democratic Sen. Kirsten Gillibrand of New york requested unanimous consent -- a procedural move that allows a bill to skip several steps to pass unanimously, without senators casting an individual vote -- on the bill on the Senate floor Wednesday so that it be accelerated to a vote without debate, Paul objected.

"It has long been my feeling that we need to address our massive debt in this country -- we have a $22 trillion debt, we're adding debt at about a trillion dollars a year," he said. "And therefore any new spending that we are approaching, any new program that's going to have the longevity of 70, 80 years, should be offset by cutting spending that's less valuable."

"We need to at the very least have this debate. I will be offering up an amendment if this bill should come to the floor, but until then I will object," added Paul, who voted in favor of President Donald Trump's $1.5 trillion tax cut. That tax cut is helping drive a deficit increase.

Paul was not the only senator who objected to the attempt to pass the bill by unanimous consent on Wednesday. Sen. Mike Lee of Utah "alerted the cloakroom that he objected to the bill passing without a vote," Lee's communications director Conn Carroll told CNN.

Though he did not object on the Senate floor in response to Gillibrand's proposal, "he is seeking a vote to ensure the fund has the proper oversight in place to prevent fraud and abuse," Carroll added.

Speaking on the floor afterward, Gillibrand said that she was "deeply disappointed" in Paul's decision.

Gillibrand grew emotional describing how first responders have "had to use their most precious commodity, time" by being away from their loved ones "to come here -- to walk the halls of Congress, to go to office after office, to ask that this body and this government stand by them in their greatest time of need."

"We could pass this bill right now," she added. "But instead, my colleague has objected, asking people to come back over and over. Everyone loves to point fingers in this place, where there's nowhere else to point that finger today than this chamber."

Senate Minority Leader Chuck Schumer also slammed Paul's objection, saying on the floor that he would ask Paul to consider the American tradition that when people "volunteered in the armed services and risked their lives for our freedom, we came back and gave them health care."

"I would urge my friend from Kentucky to withdraw his objection," Schumer added. "I would urge Sen. McConnell the leader to put it on the floor now. And we can let these folks in the gallery and so many others, do what they need to do -- help their families, help their friends, and make sure their health is given the best, best protection possible."

Paul addressed his objection later Wednesday on Twitter , arguing that he was "not blocking the 9/11 bill - simply asking for a vote on an amendment to offset the cost."

Speaking to Fox News later Wednesday, Stewart slammed Paul's objection as "absolutely outrageous," accusing the senator of "fiscal responsibility virtue signaling" and blasting Paul's support for the deficit-raising tax cut.

"Rand Paul presented tissue paper avoidance of the $1.5 trillion tax cut that added hundreds of billions of dollars to our deficit, and now he stands up at the last minute after 15 years of blood, sweat and tears from the 9/11 community to say that it's all over now, now we're going to balance the budget on the backs of the 9/11 first responder community," Stewart said.

"There are some things that they have no trouble putting on the credit card," he added. "But somehow when it comes to the 9/11 first responder community -- the cops, the firefighters, the construction workers, the volunteers, the survivors -- all of a sudden, man we've got to go through this."

John Feal, a 9/11 first responder speaking alongside Stewart, called Paul and Lee "opportunists" and "bottom feeders."

"You can't cherry pick and choose when you want to be a conservative, fiscal hawk, that's just insulting to our intelligence," Feal said. "And shame on them -- they lack humanity, they lack leadership."

In a news conference following floor proceedings, Gillibrand called on McConnell to put the bill up for a floor vote, saying, "I urge you to do the right thing" and that she appreciated his commitment to addressing the bill before Congress' August recess.

"Senator Paul may have turned his back on our first responders today, but now, we have a filibuster proof bipartisan support of 73 co-sponsors in addition to myself -- which means that there are no more excuses for those on the other side," she added.