The GOP bill would have fundamentally overhauled Medicaid from an open-ended federal guarantee to a system that caps funds to the states but would have given them more flexibility on how they spent those dollars.
House Speaker Paul Ryan said he and House Republicans were "a little frustrated the Senate has not acted on a seminal promise."
Ryan noted that his conference had done its job, passing legislation in May.
Instead, congressional leaders and the president are ready to move on to overhauling the tax code.
They are set to unveil a "framework" for their legislation Wednesday, and McConnell said the Senate Budget Committee will mark up its resolution on taxes next week.
President Trump said Tuesday he had asked members of Congress from both parties to "discuss our framework for tax cuts and tax reform before it will be released tomorrow. We will be releasing a very comprehensive, very detailed report tomorrow. And it will be a very, very powerful document."
Trump said the plan will be based on four principles:
1. "Make our tax code simple and fair." (He promised Americans would be able to file their taxes on a "single page.")
2. "Cut taxes tremendously for the middle class, not just a little bit but tremendously." (Double the standard deduction and increase the child tax credit.)
3. Lower business taxes.
4. "Bring back trillions of dollars in wealth parked overseas."
A comprehensive tax overhaul has not happened since 1986.
"Tomorrow is the beginning of a very important process that we are excited about here in Congress," Ryan said.
As for health care, McConnell tried to paint the debate as one of the Graham-Cassidy bill versus a single-payer system. Sen. Lindsey Graham, one of the bill's principal authors, had framed it as "federalism versus socialism." Independent Sen. Bernie Sanders, as well as some Democrats, have touted a "Medicare for all" plan.
But Senate Minority Leader Chuck Schumer of New York took to the Senate floor Tuesday to knock McConnell's argument as a "straw man" and a "false choice."
"Democrats have a lot of ideas about how to improve health care," Schumer said. "Each of them endeavors to increase coverage, improve the quality of care, and lower the cost of care. None — none of the Republican plans manage to achieve those goals. That's the difference. The difference is one side wants to cut health care to average Americans, increase premiums, give the insurance companies far more freedom, and one side wants to increase care, the number of people covered, lower premiums, better coverage. That's the divide."
Schumer also accused Republicans of not wanting to have that debate on the merits and called for a "bipartisan way to improve the existing system."
Later, after the announcement of the bill's demise, Schumer called for a bipartisan approach. Standing next to Washington state Democratic Sen. Patty Murray, Schumer promoted the work of Murray and Republican Lamar Alexander of Tennessee, chairman of the Health, Education, Labor and Pensions Committee, to try to fix the current system.
"I saw Sen. Alexander in the gym this morning," Schumer said, "and he seemed open to it [working on a bipartisan deal]."
Copyright 2017 NPR. To see more, visit http://www.npr.org/.