Bush made the comments in answer to questions from Catholic journalists meeting in Baltimore. He was speaking by teleconference from Austin.
The governor said he favors capital punishment as a deterrent to crime: "I believe it sends a chilling message that there is a consequence to your actions."
Texas executed 35 inmates last year, the most in the nation, and is ahead of that pace with 18 so far this year. Bush has spared only one condemned inmate in the 5 ½ years he has been governor, rejecting pleas from the Vatican, the secretary of state and officials of foreign countries whose citizens have been convicted in Texas.
In February, Karla Faye Tucker was executed despite a plea from Pope John Paul II.
The Texas governor was asked whether he had reconsidered his position in light of a moratorium ordered by Illinois' governor, a fellow Republican, who raised concerns about innocent people on death row.
George Ryan said in January that he decided to impose the moratorium after realizing that, since 1987, Illinois had released more wrongfully convicted death row inmates than it had executed.
In Texas, Bush said, "I don't believe we've executed a single innocent person. If the DNA testing helps to settle a case, or erase any doubts or concerns, we would support that."
At another point, Bush drew applause from the crowd when he spoke of his belief in the "sacredness of every human life," a reference to his stance against abortion.
Joseph Ryan, managing editor of The Catholic Standard and Times in Philadelphia and outgoing president of the Catholic Press Association, said the church is not just opposed to abortion but "is pro-life through the entire continuity of life."
The church is opposed to the death penalty in most cases, but many Catholics favor capital punishment, Ryan said. As to how that plays in elections, he added, "Catholics have been kind of split across both parties."
Bush was criticized during the Republican primaries after speaking in February at Bob Jones University, a South Carolina school whose leaders have espoused anti-Catholic views. He later apologized for not being "more clear in dissociating myself from anti-Catholic sentiments and racial prejudice."
The newspapers represented by the group reach 20 million of the nation's more than 60 million Catholics, who account for more than a quarter of the electorate, CPA officials said.
During his prepared remarks, Bush said the Vatican should be allowed to maintain its "permanent observer status" at the United Nations, and he said the Clinton administration's position was unclear.
State Department spokesoman Susan Elbow, asked for a response, said the United States "thinks the Holy See has a positive role to play as an observer at the U.N. and would not support its removal, or any efforts to downgrade its status."
And Gore campaign spokesman Chris Lehane said: "Once again the governor is wrong. Al Gore has long supported the Vatican maintaining its permanent observer status within the U.N."
The Vatican, classified as a nonmember state with permanent observer status, has many of the same rights and privileges as the 188 countries that are full U.N. members. Though it has no vote, the Vatican can take active part in debates and conferences, where it has made a strong mark on family planning and abortion issues.
The private group Catholics for a Free Choice, which supports abortion rights, is seeking a formal review of the Holy See's status, saying the Vatican's claim to be a state is questionable.