The reality television show host also insinuated that in the past, the United States would have attempted to covertly kill someone leaking information like Snowden.
"You know what we used to do to traitors, right?" Trump said.
"We can't allow this guy to go out there and give out all our secrets and also embarrass us at every level," he added.
Trump said he supported efforts by the Obama administration to extradite Snowden back to the United States, including Vice President Biden's call to Ecuador asking the country not to grant Snowden asylum, adding that he did not believe "it makes us look weak at all." But, Trump said, the administration should have asked Ecuador to trick Snowden into believing he had been granted asylum, and then turn him back over the United States.
Trump said that he did not believe Russian authorities would ever turn Snowden over. The former defense contractor is currently at the Moscow airport.
"Putin is so much stronger than our guys, it's almost an embarrassment," Trump said.
Trump, who flirted with a presidential run during the 2012 Republican primaries, also sought to downplay a trip to Iowa planned for next month. He said he was making the trip "for friendship," not presidential ambitions.