Alcaraz needed medical attention after twisting his right ankle on the second point of the contest, but the Spaniard returned to the court a little later with heavy strapping.
He played through the pain and managed to break Brazilian Monteiro's serve but retired after being broken in the next game as the match ended with the score at 1-1.
"Tomorrow I'll have a test for my ankle and let's see if it's something serious or not," Alcaraz told reporters.
"I felt bad. That was the first impression I had. I was feeling pain once I fell down, so I thought it was going to be difficult to continue if I was still having those feelings.
"After a few points, it didn't feel better. I couldn't move well and I knew that it was going to be impossible to continue. I thought it was going to get worse if I kept playing for such a long match and that's why I choose to retire."
Top seed Alcaraz won the Rio title in 2022 and was runner-up last year, when he lost to Briton Cameron Norrie. He arrived in Rio after losing to Chilean Nicolas Jarry in the semi-finals of last week's Buenos Aires Open.
Alcaraz announced earlier on Tuesday that he would play at the Queen's Club Championships before defending his Wimbledon title in July.
The twice-major champion will hope to be fit enough to defend his Indian Wells title next month.
