It was the first in a series of politically tricky amendments on the Iran deal.
The freshman Arkansas Republican is a first-time dad.
"I would just urge all people to exercise restraint," Ben Cardin said.
"It shouldn't be up to Democrats to kill these amendments," he said.
Sen. Ben Cardin (D-Md.) said he and Sen. Bob Corker (R-Tenn.) are looking for an "orderly way" to tackle amendments.
More than a dozen senators missed the vote.
Sens. Ben Cardin and Bob Corker are hoping to move forward with the legislation.
The Senate Democratic leader said the law had turned out to be a "smashing success."
New amendments set up hurdles as lawmakers work on a path to passage.
The senators said the VA has the authority to grant benefits to C-123 veterans.