The summits are viewed as symbols of peace on the Korean Peninsula, as statements issued following the meetings were aimed at building cooperation between the two Koreas.
He also reiterated the Trump administration's stance that the United States would take whatever "action is necessary to defend our homeland", including military operations, as it seeks to denuclearize the North.
A White House official said that, although Moon did not discuss the invitation with Pence on Saturday, the South Korean president made it very clear that only when North Korea actually starts to take steps to denuclearise would anyone even consider beginning to take the pressure off.
Moon's desire to engage North Korea was in contrast to his U.S. ally.
Pence's staff insists the vice president would have met with the North Korean officials if the North Korean officials were interested in meeting.
Direct negotiations between North Korea and the USA took place before the two previous inter-Korean summits.
Moon shook hands with Kim's sister at the presidential Blue House in Seoul. I walk into the locker room and you can't tell who is from the North and who is from the South. North Korean state media said Thursday there was no intention on the North Korean side for such talks to take place.
But if Pyongyang plays ball, the Trump administration will be put to an unavoidable choice between sitting down with the Kim regime under what are sure to be imperfect circumstances or sticking to its demands and further alienating the Moon government.
Past administrations have been wary of efforts by Pyongyang to drive a wedge between Washington and Seoul, but still generally supportive of efforts to calm tensions at the heavily militarized border that divides the Koreas.
The North's participation has been generally welcomed, but right-wing protesters have shown up at several venues to burn North Korean flags and tear up portraits of Kim's brother.
"The girls have been great; they fit in well with our team", said Marissa Brandt, who was adopted as an infant out of South Korea and whose sister plays on the USA team.
But both South Korea and the United States has been cautious about the talks, saying there will not be progress without North Korea giving up nuclear weapons.
But both sides have held their ground.
From the moment they entered the stadium to the tune of "Eye of the Tiger" and Darude's "Sandstorm", the North's cheerleaders knew they had competition.
The allies appeared to agree that North Korea should make progress on denuclearization before a summit can be held.
The Perry Process included freezing North Korea's nuclear and missile programmes and normalising relations between Pyongyang and Washington.
Some analysts were skeptical of that prospect. Kim's sister, Kim Yo-jong, personally delivered the message as she was in Seoul for the Winter Olympics. "With his summit proposal, he seeks to incite friction between Seoul and Washington by widening their policy gap".