He also said the deal will give Democratic leaders more than enough votes to pass the bill.
"We're well past" the 216 votes needed for passage, Stupak said. "This bill is going to go through."
President Barack Obama will issue an executive order "after the passage of the health insurance reform law that will reaffirm its consistency with long-standing restrictions on the use of federal funds for abortion," White House Communications Director Dan Pfeiffer said in a written statement.
"While the legislation as written maintains current law, the executive order provides additional safeguards to ensure that the status quo is upheld and enforced, and that the health care legislation's restrictions against the public funding of abortions cannot be circumvented."
Related: Read the entire order
Under the bill now being considered by the House of Representatives, federally-funded abortion coverage for people purchasing insurance through new insurance exchanges would be banned. Exceptions would be made in cases of rape, incest, or danger to the life of the mother.
Individuals receiving federal assistance who want abortion coverage would have to purchase the coverage using private funds.
Anti-abortion rights Democrats successfully defended a "principle that meant more to us than anything, and that's the sanctity of life," Stupak said.
Stupak said he had been reassured that Obama's order will ensure the health care bill complies with the 32-year-old Hyde Amendment, which sharply restricts federal funding for abortion.
"This legislation is a whole-life piece of legislation," said Rep. Kathy Dahlkemper, D-Pennsylvania, an abortion rights opponent who backed the Stupak deal.
GOP leaders argued the deal would do little to ensure taxpayer dollars are not used to fund abortions.
"The law of the land trumps any executive order, which can be reversed or altered at the stroke of a pen by this or any subsequent president without any congressional approval or notice," said House Minority Leader John Boehner, R-Ohio.
"Make no mistake, a 'yes' vote on the Democrats' health care bill is a vote for taxpayer-funded abortions."
Updated: 5:00 p.m.