Just weeks on the job, Henry Gomez, the new president of the U.S. business for Skype Technologies, is using the Consumer Electronics Show this week to get his head around the exploding consumer market for Web-based phone calling.

Skype, which has helped popularize Internet-based phone calling, is already an international sensation, attracting millions of daily users.

Gomez, the former vice president of communications at eBay, has been charged with making quick in-roads into the United States market, where the two-year-old Skype is only beginning to become better known among mainstream Web users.

Skype must contend with a host of potential competitors from big phone companies to major Internet rivals of its parent company, eBay. Gomez, who also handled governmental relations for eBay, will have to gear up for likely regulatory battles meant to hobble its free or low-cost phone calling services.

“My goal is to make Skype a household name,” Gomez said, comparable to Europe where “Skype me” has become a trendy way for saying “call me.”

He speaks with amazement of the way people are finding new uses for Skype — from non-native English speakers finding people on Skype to help them with their accents to law students who no longer need to meet face to face for case study sessions. Instead classmates show up at an appointed hour on Skype for group video conference calls.

His favorite Skype anecdote so far is the Skype baby monitor. According to Gomez, one U.S. father who spends a lot of his time traveling in Asia has set up a computer in his baby’s room using Skype to act as a remote baby monitor. While his wife sleeps, the father works at his computer in Asia, casually listening if one of his children should awake in the middle of the night.

Throughout the night, the father is there to reassure his kids. Of course he’s half a world away but in constant virtual contact.

Could nannies or babysitters fall victim to this dangerous new outsourcing trend?