"It's that important to me," he told CNN. "This is a road we haven't gone down before. If you can't show the American people that international organizations can be more responsible, there is going to be a break. And I am going to lead that break."
"I will do everything in my power, working with the new administration and Congress, to leave no doubt about where America stands when it comes to the peace process and where we stand with the only true democracy in the Middle East, Israel," Graham added. He later told CNN's Dana Bash that US funding accounts for 22% of the UN's budget.
The United States on Friday allowed a UN Security Council resolution condemning Israeli settlement construction to be adopted, defying significant pressure from Prime Minister Benjamin Netanyahu's government in alliance with President-elect Donald Trump.
The Security Council approved the resolution with 14 votes, with the US abstaining.
Trump soon after tweeted, "things will be different after Jan. 20th," and on Friday, a senior Trump transition official told CNN the transition team was "all hands on deck" in working to kill the UN resolution.
Graham issued a statement after the vote, harshly criticizing the Obama administration for abstaining.
"The Obama-Kerry foreign policy has gone from naïve and foolish to flat-out reckless. With friends like these, Israel doesn't need any enemies," he said. "I anticipate this vote will create a backlash in Congress against the United Nations. The organization is increasingly viewed as anti-Semitic and seems to have lost all sense of proportionality."
Graham traveled to New York last week to discuss the resolution with António Guterres, the next UN Secretary General, and he spent the past week lobbying members of the Security Council not to support the UN vote.
The South Carolina Republican also talked to South Carolina Gov. Nikki Haley, Trump's pick for ambassador to the United Nations, about the resolution.
"The settlement issue is no doubt an issue," Graham told CNN. "But this is not the way. We abandoned Israel."
Citing the controversial 2009 Goldstone report about the Israeli conflict in Gaza, which accused both Israel and Hamas of war crimes, as well as other anti-Israel measures at the UN, Graham said, "enough is enough."
"The bottom line is with all of the problems we have, Israeli settlements are pretty low on my list." Graham said.
He added that he wanted to work with the UN to help stabilize Iraq and Syria. He also was planning a join efforts with the UN to create a Marshall Plan to support countries emerging from conflict, but won't do it unless the UN overturns the resolution
"I am a huge supporter of foreign aid and the UN," he told CNN. "I want to do more, not less. But I can't support funding a body that singles out the only democracy in the Middle East who shares our values."
Graham plans to move forward with his measure after the congressional recess when a continuing resolution expires in April.
Senator Ted Cruz tweeted that he spoke with Netanyahu and assured him of support for Israel in Congress. "No US $ for UN until reversed," Cruz wrote on Saturday.
Graham hopes to create leverage for Trump to deal with the United Nations on the issue when the President-elect takes office.
"Trump is a good negotiator. Let's see if this give him some leverage," he said, adding if the world body overturned the measure, the UN "has got a chance to reset and let the new administration restart the peace process."