The Roman Catholic Archdiocese of Indianapolis also said Tuesday it has brought a Syrian refugee family into the state despite Republican Gov. Mike Pence's request that it not do so.
Archbishop Joseph Tobin said the family arrived safely in Indianapolis Monday night.
Tobin met last week with Pence to discuss the governor's order blocking state agencies from assisting Syrian refugees following the deadly Paris attacks last month. Pence said he asked Tobin to not resettle the family in the state.
The resistance to Syrian refugees comes amid heated anti-Muslim rhetoric on the national political stage by some in the Republican party.
Donald Trump recently called for a halt to all Muslims entering the United States, and later defended his plan by comparing it to President Franklin D. Roosevelt's Japanese internment camps during World War II.
Sen. Ted Cruz, R-Tex., who is currently leading 2016 polls in Iowa, joined Texas Gov. Greg Abbott on Tuesday in calling for a halt to all refugees coming from any country where the Islamic State of Iraq and Syria is active, reports CBS Dallas.
"The FBI has told us they don't have the resources to properly vet those refugees to determine whether or not they are ISIS terrorists," Cruz said.
Tobin said in a statement released Tuesday morning that he "prayerfully considered" Pence's request but decided that assisting the refugee couple with two small children was an "essential part" of the Catholic Church's identity.
"Three years ago, this family fled the violence of terrorists in their homeland of Syria. After two years of extensive security checks and personal interviews, the United States government approved them to enter our country," Tobin said.
In a statement, the governor's office said: "The governor holds Catholic Charities in the highest regard but respectfully disagrees with their decision to place a Syrian refugee family in Indiana at this time."
An aide later said Pence wants residents of the state to welcome a Syrian refugee family that was settled there, despite his objections. Pence spokesman Matt Lloyd said Tuesday that the Republican governor is confident that Indiana residents will be welcoming to the family.
The governor cited again concerns that terrorists were slipping into the country with refugees, and said the Obama administration in 2009 "paused the Iraqi refugee program for six months after the FBI found evidence that terrorists from Iraq infiltrated the United States through the refugee program in Bowling Green, KY."
Initially, it appeared Pence's order might hold up, despite its lack of legality.
In November, a Syrian family originally bound for Indiana was diverted to Connecticut. Exodus Refugee Immigration Inc. said at the time they did not believe the state has the power to stop the resettlements but didn't want the family to go through any additional "scrutiny or drama."
The family of three had been waiting since 2012 to be resettled in the U.S.
Also in Texas, six Syrian refugees have settled in their Dallas home over the initial objections of Texas officials who wanted to stop them.
International Rescue Committee spokeswoman Lucy Carrigan said Tuesday that the family arrived Monday afternoon. She described the family as relieved and eager to start a new life in the United States.
Carrigan declined to identify where the family was living due to safety concerns.
The family's arrival was closely watched because it comes after Texas officials said they wanted to ban new Syrian refugees following the deadly Nov. 13 Paris attacks.
When the IRC indicated it would continue with resettling the family and other Syrians in Texas, the state went to court. The state has since withdrawn its request for an immediate ban on Syrian arrivals.