Some held signs that read, "Not trivial," in response to Christie's veto message that called the proposed reduction of the gun magazine capacity limit from 15 rounds to 10 a "trivial approach" that would not reduce mass violence.
Of the Newtown families, Christie said it was "certainly their prerogative" to disagree with him and to express their views.
While "I have nothing but sympathy for those folks . . . I don't believe that the bill that was passed in New Jersey was an effective way to deal with it," Christie told reporters in the Glory Days Diner, where he campaigned with Republican candidate Tom Foley in his role as chairman of the RGA.
A Greenwich businessman who was named ambassador to Ireland by President George W. Bush, Foley ran for governor in 2010 and lost to Malloy by just 6,400 votes. Christie endorsed Foley in 2010.
In August, Foley will compete in a contested Republican primary with state Senate Minority Leader John McKinney. He has led McKinney by a wide margin in recent polls.
Christie said it was "a hard decision to make" to involve the RGA in a contested primary, a step he also took while visiting New Hampshire in June.
Christie said that Foley earned his confidence four years ago, and noted that party delegates endorsed Foley during a state convention this year.
Jodi Latina, a spokeswoman for McKinney, declined to comment Monday on Christie's appearance with Foley.
Foley has been polling neck-and-neck with Malloy, a vocal opponent of Christie's. The two have criticized each other publicly. On Monday, Christie said, "I know Gov. Malloy's thrilled I'm here today."
Christie said the RGA was committed to winning the governorship in Connecticut, which has a largely Democratic electorate.
"I know, as a guy who won twice in a blue state, these are winnable races," Christie said.
Asked whether he would campaign for New York Republican Rob Astorino against Democratic Gov. Andrew Cuomo - who has a big lead in polls - Christie said the RGA does not "invest in lost causes."
In endorsing Foley, Christie aligned himself with a Republican candidate seen as a Second Amendment supporter, said Gary Rose, a political science professor at Sacred Heart University in Fairfield. McKinney voted for the gun-control bill the Connecticut legislature passed last year.
Foley would not tell reporters Monday whether he supported Christie's magazine capacity veto, saying he was focused on economic issues.
It also does not hurt Christie - who was also in Greenwich to raise money for the RGA - to meet wealthy donors, Rose said.
"It is true that presidential candidates head straight to Fairfield County," one of the wealthiest counties in the country, Rose said.
Before addressing reporters in the diner, Christie spent about 15 minutes shaking hands, posing for photos, and planting a kiss on one woman's cheek.
As he left, Louise Bavis stood watching, a smile on her face. "I like him," said Bavis, a retired secretary and member of the Greenwich Republican Committee. "He seems like he likes people a lot."
Less impressed was Richard Boritz, who asked Christie in the diner about his veto of the magazine capacity bill.
"He gave me the pat answer, which is, it's a mental-health issue," said Boritz, a retired school librarian who moved from Newtown to a nearby town two months ago. "It's more than a mental-health issue."
Animosity toward Christie's gun-measure veto was on full display en route to his next stop, a private fund-raiser for the RGA. About 150 protesters from Connecticut and New Jersey lined a residential street.
"Gov. Christie ought to know this is going to be an issue in presidential primaries as well," said Beth Boyer, 48, of Greenwich, who was holding a sign that read, "Republicans for Gun Control."
Across the street were about a dozen counterprotesters, including Bill Magyari, 44, of Stamford, who brought a sign that read, "Thank you Chris Christie for rational thinking."
Magyari said he would "possibly" vote for Christie in 2016, but said he would have ruled the New Jersey governor out if the gun magazine bill had become law. "He's kept me on the hook because of the veto," Magyari said.