Rodriguez's next court date is August 1.
The 39-year-old is charged with manslaughter and criminally negligent homicide. The siblings were pronounced dead at the scene.
He drove to work at the James J. Peters VA Medical Center in the Bronx, parked and went inside, leaving the babies behind.
"He carried on with his day", the New York Post reported that Assistant District Attorney Jaime Breslin told the judge at Rodriguez's arraignment.
"He forgot his children in the seats".
When he returned to his auto eight hours later to drive home, he "observed Luna and Phoenix in the above reference vehicle, strapped into their auto seats, not moving, not breathing and both children appeared lifeless", according to court documents.
He was released on $100k (£81.5k) bond that day, according to the New York Post.
Rodriguez told police, "I blanked out".
In a statement to NBC, she said: "I will never get over this loss and I know he will never forgive himself for this mistake".
"Although it hurts more than I could ever imagine, I love my husband still".
Marissa Rodriguez says her husband, Juan Rodriguez, is a good person and would never have hurt the children intentionally.
A NY father who was charged in the deaths of his twin babies is "devastated", according to his defense lawyer.
The family is "trying to come to grips with the horrific nature of this circumstance", Jackson said.
Juan Rodriguez, 39, told investigators he dropped off his 4-year-old son at one day care provider Friday, then forgot to drop the 1-year-olds at their day care location.
One neighbour said: "He was always doing things with the kids".
There have been 23 deaths of children left in hot cars so far this year. "Always. This is just a frightful situation", said Temple Barros, 41. "Always. This is just a awful situation", he said. "He would always play catch or basketball with his older kid". "No one here can believe it".