The bill, a package containing some $85 billion in more than 50 tax cuts, failed 53-40 on a cloture vote that would have required 60 votes to advance.
Although many Republicans support the measure, they blocked cloture after Reid refused to allow amendments on the medical device tax that was part of the Affordable Care Act.
"What are they afraid of?" McConnell charged. "Now they want to change the Bill of Rights. Apparently after a couple hundred years they've decided that Madison's work is not sufficient."
Reid and McConnell have frequently sparred over the use of amendments, with each side accusing the other of abusing the process to block legislative progress.
"We have a tax bill here that members from both sides want to improve and support," McConnell said.
"Yet we don't get a chance to amend it. [Democrats] have turned the Senate into a graveyard of good ideas and good democratic debate."
But Reid said it was his GOP colleague who was being unreasonable, saying he had offered to allow the bill's author, Finance Committee Chair Ron Wyden, D-Ore., and ranking member Orrin Hatch, R-Utah, to try to hammer out a deal once cloture was invoked.
"I've already said we will be allowed to have amendments but [they] have to be related to the bill, but [Republicans] want to offer them on everything," Reid said. "I think it's just outrageous that they are going to try to kill thing on this phony issue."
The breakdown of the tax bill was the second time in a week partisan squabbling has gotten in the way of passing widely supported legislation. On Monday, a similar fight over amendments sunk a deal to pass both an energy efficiency bill and the Keystone XL pipeline bill.