The Democratic presidential front-runner has been hammering at the Bush administration for two weeks since a top Pentagon aide refused to tell her whether or how the military was planning for the eventual withdrawal of U.S. forces from Iraq.
In a letter to the vice president, she accused Cheney of offering "inaccuracies" in opposing her request.
She used even tougher language in an appeal sent to supporters of her presidential bid: "I couldn't care less what Dick Cheney says about me. But when he plays politics with the lives of our troops, you had better be sure I'm going to respond. And I know that you want to respond too."
The politically charged debate began in May when Clinton, a member of the Senate Armed Services Committee, sought a briefing on how the Pentagon is planning to eventually remove troops and equipment from Iraq.
Undersecretary for Policy Eric Edelman wrote to the New York senator last month that such discussion boosts "enemy propaganda." Clinton called that answer "outrageous" and accused the Pentagon of ducking a serious issue.
Now she's taking aim at Cheney, who said in an interview Tuesday with CNN that he agreed with Edelman's letter. Cheney said discussing specific troop movement plans with Congress could tip off the enemy.
"I thought it was a good letter," Cheney said.
Clinton fired back Wednesday in the letter to Cheney, saying she had never sought specifics of troop movements and was, in fact, getting a briefing from Edelman on the subject Thursday.
"I feel it is necessary to respond to several comments and inaccuracies you put forward," Clinton wrote, noting that she had also received a conciliatory letter from Defense Secretary Robert Gates.
"Your comments ... have left me wondering about the true position of the administration. Therefore, I am writing to President Bush asking that he set the record straight about the administration's position regarding the role of Congress in oversight of the war," she said.
A public feud with the Pentagon generally and Cheney in particular may help Clinton among Democratic primary voters still uneasy over her 2002 vote to authorize the invasion of Iraq.
Her two main rivals, Sen. Barack Obama and former Sen. John Edwards, have sought to position themselves as more consistently and more stridently anti-war than Clinton.
The Clinton campaign issued an appeal from the senator to supporters, asking them to weigh in on the issue of Cheney and troop withdrawal planning.