Trump made the last-minute decision Thursday night after the downing of a Navy drone that Washington said was flying within international airspace.
The mission would have reportedly targeted Iranian missile batteries and radars.
Trump took to Twitter Friday morning to confirm he called off the strikes and said he based his decision on the number of potential casualties.
A source had confirmed to Fox News earlier Friday morning that the administration made a last-minute decision to call off the retaliatory strikes against Iran.
But until Trump's tweets, few details about the aborted mission and the circumstances that led to the reversal were publicly available. And it remained unclear Friday whether strike plans have been definitively shelved.
Responding on "America's Newsroom," Wallace questioned Trump's explanation that potential casualties were the reason for his decision. Wallace said such discussions about "collateral damage" would normally occur far in advance of the president ordering the strike to go forward earlier on Thursday.
"A war with Iran would be a very ugly and prolonged conflict so it does raise the question: Does the president really have the stomach to launch this kind of attack and get more deeply involved, perhaps, in a confrontation with Iran?" Wallace said.
He stated Iran bears the responsibility for provoking the confrontation, but there are costs to both action and inaction.
"Remember when President Obama decided a few years ago, he declared that red line and said we would strike back if Syria launched another chemical weapons attack and then he backed off that. A lot of people, including Donald Trump as a candidate and even as president, said there was a real cost to that in terms of the U.S. threatening something and then not going ahead and doing it," Wallace recalled.
He concluded Trump may still order a strike with less of a risk of causing casualties, but it appears the president "doesn't have much of an appetite" for such an escalation.
Fox News' Lucas Tomlinson contributed to this report.