Federal officials claim the migrants may not have died if they were not barred from conducting their patrols by Texas National Guardsmen.
Mr Abbott directed the guardsmen to seize a park in the border town of Eagle Pass, Texas, where the US Border Patrol had been processing migrants who crossed into the US. The guardsmen were instructed to refuse entry to federal officials who patrolled the border in the area.
Mexico's National Institute of Migration identified the migrants who died on Monday. The victims are Victerma de la Sancha Cerros, 33; Yorlei Rubi, 10; and Jonathan Agustín Briones de la Sancha, 8, according to CNN, citing the Mexican agency.
Democratic Congressman Henry Cuellar of Texas said the migrants were a mother and her children, according to Mother Jones.
Mr Abbott passed a law last month that made illegal migrant crossings into Texas a state crime, which he argues allows state law enforcement the right to arrest undocumented migrants anywhere in the state.
Federal officials have maintained that enforcing immigration is solely done by the US government, not the state.
Shelby Park in Eagle Pass has been a site of frequent friction between Mr Abbott and the federal government. The state previously seized the park after the town's mayor, under the direction of the Texas Department of Public Safety, claimed it as private property.
Mr Abbott used the park as an operations centre for his "Operation Lone Star," which uses Texas National Guard troops to enforce illegal immigration into the state.
The programme was criticised after four guardsmen forced to participate in the operation died by suicide and for the construction of a barrier in the Rio Grande that opponents said threatened both the safety of migrants and the environment.
On Saturday, Mr Cueller said that Texas officials refused to allow US Border agents access to the site to save the drowning migrants. However, a filing by the DOJ with the US Supreme Court has refuted those claims, stating that Border agents were not notified about the drownings until after they occurred.
However, the DOJ filing does note that two other migrants were "in distress" on the US side of the Rio Grande in a separate incident and that guardsmen physically barred US Border agents from assisting the migrants. Those migrants were eventually saved by Mexican law enforcement.
The filing claims that “it is impossible to say what might have happened if Border Patrol had had its former access to the area…at the very least, however, Border Patrol would have had the opportunity to take any available steps to fulfill its responsibilities and assist its counterparts in the Mexican government with undertaking the rescue mission. Texas made that impossible”.
“Texas stands in the way of Border Patrol patrolling the border, identifying and reaching any migrants in distress, securing those migrants, and even accessing any wire that it may need to cut or move to fulfill its responsibilities," the DOJ said in its filing.
The Biden Administration responded to the seizure of the park by sending a cease and desist letter to Texas Attorney General Ken Paxton, claiming Texas's actions were unconstitutional, according to the Texas Tribune.
“The Texas governor’s policies are cruel, dangerous, and inhumane, and Texas’s blatant disregard for federal authority over immigration poses grave risks,” the US Department of Homeland Security said in a statement reported by Mother Jones.
The federal government has given the state until Wednesday to remove its forces from the border.