When asked whether the secretary general had any contact with Kim Jong Un's sister, Kim Yo-jong, the spokesman replied that their direct communication was unlikely.
Earlier on Friday, Pence said Moon gave his backing to additional measures the United States is planning to try to curb North Korea's missile and nuclear programs, speaking after paying tribute at a South Korean memorial.
Moon met Kim Yo Jong and the North's ceremonial head of state, the elderly Kim Yong Nam - technically the highest-level Northern official ever to go to the South - for talks and lunch at the Blue House on Saturday in a landmark meeting.
After the game, Moon and the two Kims came down to the Korean team bench to speak with the players gathered on the ice as the cheerleaders sang in the background.
"We are a sad, divided nation and when it comes to solving this problem, there is no South or North Korea", said Jo Jung-rae, one of South Korea's most well-known novelists, in an interview last week.
US Vice President Mike Pence, sitting a few seats away in the same Opening Ceremony VIP booth, looked stony faced as the scene unfolded. The sight of North and South Koreans marching together for the first time in a decade was the culmination of months of work by Seoul.
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Officials in Tokyo suspect North Korea is trying to chip away at the united front of the United States and its East Asian allies as global sanctions bite the hermit country.
The missive was delivered by Kim's visiting sister Kim Yo-jong, who said he was willing to meet "at the earliest date possible", a spokesman for South Korea's presidential palace said.
Saturday's meeting also involved two other high-level delegates - Choe Hwi, the chairman of the National Sports Guidance Committee, and Ri Son-gwon, the head of the North's state agency in charge of inter-Korean affairs.
Kim Yong Nam and Kim Yo-jong - were also in the box, seated in a row behind Pence.
His office said he did not interact with the North Koreans.
"Working in unity on the way to unification", Kim Yong Nam wrote in the guest book.
She is a senior member of the North's ruling Workers' Party and a key adviser to her brother.
Switzerland's Alina Muller nearly single-handedly spoiled Korea's Olympic unity moment on Saturday, scoring four goals to power the Swiss to an 8-0 win over Korea's unified women's ice hockey team.