The shooting occurred at around 11 a.m. local time on Saturday (16:00 GMT).
"The Methodist Health System Family is heartbroken at the loss of two of our beloved team members," the system's executive leadership said in a written statement quoted by NBC on Saturday.
Charley Wilkison, the CEO of the Combined Law Enforcement Associations of Texas (CLEAT), said on social media that the suspect shot two maternity ward nurses.
"111th @CLEAT Critical Incident, OIS [officer-involved shooting], Dallas Methodist Hospital. Suspect shot two maternity ward nurses. Officer heard shots, saw suspect w/gun, fired one round, suspect hit. Officer OK! Staff Atty Ben Casey responded to OIS," Wilkison said.
A Methodist Health System police officer confronted and shot the suspect, who is now in custody, according to hospital officials.
"The suspect was detained, stabilized, and taken to another local hospital," NBC quoted Methodist officials as saying on Saturday.
The motive behind the shooting remains unclear.