"Of course I'm sorry about a botched joke. You think I love botched jokes?" Kerry said during an appearance on a US radio programme.
"I mean, you know, it's pretty stupid."
Kerry, who lost the 2004 presidential election to Bush, on Tuesday also cancelled several appearances he was to make with other Democrats this week ahead of the country's mid-term elections on November 7.
The political furore erupted following Kerry's speech to a group of California students on Monday in which he said people who do not study hard and do their homework would likely "get stuck in Iraq".
Kerry's aides later said he fumbled the delivery of a line aimed at Bush which was originally written to say "you end up getting us stuck in a war in Iraq".
'Smear and fear'
Kerry has since said he meant no offense to troops, countering that the White House had twisted his words and that Bush owed the troops an apology for a misguided war in Iraq.
"I'm sorry that that's happened," he said of his comment, "but I'm not going to stand back from the reality here, which is, they're trying to change the subject.
"He's insisting on pointing fingers at the president - just say you're sorry, it's not hard"
White House spokesman
"It's their campaign of smear and fear."
However, Republicans seized on the remark, saying Kerry had insulted US troops, with George Bush, the president, calling it "insulting and shameful".
Tony Snow, the White House spokesman, said that Kerry should say sorry.
"He's insisting on pointing fingers at the president. Just say you're sorry. It's not hard," he said.
Several Democrats have since distanced themselves from Kerry's remarks.
"Whatever the intent, Senator Kerry was wrong to say what he said," said Democratic congressman Harold Ford Jr, running for the Senate in Tennessee.
But others accused the Republicans of electioneering ahead of the mid-terms.
"Instead of going on television attacking John Kerry and everyone else under the sun, the president ought to be sitting at his desk coming up with a plan for Iraq," said Senator Chuck Schumer, head of the Democratic campaign effort.