Some benedictions bestow a bigger blessing than others, and Mitt Romney hit the jackpot by signing up Cardinal Timothy M. Dolan, president of the Conference of Catholic Bishops, to deliver the closing prayer at the Republican National Convention in Tampa, Fla., next week.

The move is perhaps the clearest sign of a more aggressive push by Republicans to win over Roman Catholic voters this election cycle. Four years ago, Barack Obama won the votes of a majority of Catholics, assisted in part by his Catholic running mate, Joe Biden.

But this time the Republicans sense an opportunity to cut into that advantage, at least with moderate and independent Catholics, because of two recent developments: Obama’s endorsement of same-sex marriage and the standoff between the Obama administration and Catholic bishops over the requirement that Catholic hospitals and schools must provide coverage for birth control in their employee health insurance plans. The issue has become the centerpiece of a campaign by the bishops to defend what they consider a matter of religious liberty.

The Obama campaign is aggressively contending for Catholic voters too, and the rivalry has been sharpened this year because Catholics are in the vice presidential slot on both tickets. Biden and the Republican challenger, Rep. Paul D. Ryan, represent competing factions within their own church: Biden places a premium on the church’s social justice tradition of caring for the needy, while Ryan champions church teaching against abortion and same-sex marriage. Both are accused by their critics of being “cafeteria Catholics” willing to abandon the church teachings they do not politically support.

Romney had been courting Dolan since April. That month, the two had a private meeting, previously undisclosed, at the chancery in New York, across from St. Patrick’s Cathedral, said Peter G. Flaherty, a senior adviser to the Romney campaign who is Catholic and who served as Romney’s liaison to the religious community when he was governor of Massachusetts.

“We’re going to have outreach to Catholics in a coordinated, organized effort — state by state, diocese by diocese, parish by parish and pew by pew,” Flaherty said in an interview.

He added that Romney, a Mormon, had close ties with Cardinal Sean P. O’Malley of Boston and traveled to Rome for the ceremony at which the prelate was made a cardinal. But Romney sought out Dolan to give the benediction at the convention, Flaherty said, because of his stature as president of the bishops’ conference and his proclamations that religious liberty is at risk because of Obama administration policies. Romney has echoed this theme on the campaign trail and in a television advertisement.

It is also the message of a slickly produced video aimed at religious voters that the Romney campaign says it did not produce but that clearly helps its cause. The video juxtaposes people entering a voting booth with a blacksmith forging the words “life,” “marriage” and “freedom” in a raging fire, and it ends by asking, “Will you vote the values that will stand the test of fire?” There are two versions of the video, one for Catholics and one for evangelicals. The evangelical version has been viewed fewer than 30,000 times, while the Catholic version has been seen more than 1.9 million times.

Catholics make up about a quarter of the electorate, but they hardly vote as a bloc any longer. The Catholic vote is instead a bellwether that mirrors the general electorate. Surveys showed that in 2008 Obama prevailed among Catholic voters by 9 percentage points.

This time, the most recent poll from Gallup, taken from July 30 to Aug. 19, showed Romney with a slight edge among registered voters who are Catholic.

“Since 1972, the candidate who has won the Catholic vote has won the popular vote as well,” said Robert P. Jones, chief executive and founder of the Public Religion Research Institute in Washington. “The Catholic vote does tend to be on the side of the winning candidate. It’s the quintessential religious swing group.”

But Jones said he did not see evidence of major shifts among Catholic voters this time — not among white Catholics, who tend to favor the Republican, and not among Latino Catholics, who tend to favor the Democrat. Neither the bishops’ religious liberty campaign nor the president’s initiative to lift some restrictions for illegal immigrants who came to the United States as children has resulted in much of a change, Jones said.

Both campaigns are organizing Catholic leadership teams in the states, and they have announced slates of prominent Catholics to serve as national co-chairmen and surrogates. The Romney campaign has enlisted six former U.S. ambassadors to the Holy See as national co-chairmen of Catholics for Romney.

The co-chairmen for Catholics for Obama are scholars, prominent laypeople like Victoria Reggie Kennedy, the widow of Sen. Ted Kennedy, and politicians like Gov. Martin O’Malley of Maryland, Rep. Timothy Ryan of Ohio and Sen. Dick Durbin of Illinois.

The Obama campaign is counting on the social justice Catholics who are concerned that the Romney-Ryan team would cut the safety net for the needy and the elderly. Polls have also shown that many Catholics support same-sex marriage and do not accept the bishops’ claims that religious liberty is at risk.

Staff members with the Obama campaign who did not want to be quoted said it would be highlighting the fact that the bishops had repeatedly issued letters criticizing the budget proposed by Ryan over its failure to protect what the Bible calls “the least of these.”

With the announcement of Dolan’s role at the convention, the criticism that the bishops are turning the church into “the Republican Party at prayer” has intensified.

The invitation to Dolan circumvented church protocol, said the cardinal’s spokesman, Joseph Zwilling. Usually, the local bishop gives the prayer at the convention. So when the Romney campaign asked Dolan to deliver the benediction, the cardinal first checked with Bishop Robert Lynch of St. Petersburg, Fla., who gave his approval, Zwilling said.

Dolan accepted but said that he would be giving only a prayer, not an endorsement, Zwilling said. The cardinal also said that he had informed the Democratic Party that he would accept a similar invitation from it. A spokesman for the Obama campaign said that it had not offered one but suggested that it was close to announcing its religious lineup.

© 2012 The New York Times Company