The Premier League champions drew 1-1 in the fourth-round tie on Tuesday, then lost 5-3 on penalties. It ranks among the most embarrassing defeats in their history. Dowman saw red five minutes before the end of normal time for aiming a kick at Harrison Neal.
The decision looked harsh, but Arteta gave the midfielder a stern word as he trudged off. Some pundits said the manager should have shown more sympathy.
Arsenal lost an appeal, so Dowman now serves an immediate three-game ban for violent conduct.
Arteta stood by his response at a news conference Friday, ahead of Arsenal's trip to Liverpool. "I understand. But you need to understand the context, the relationship that I have with him and what we discussed just before we went out," he said. He asked what people expect when teammates have to play with 10, then added, "This is the environment, and this is the reality."
For Dowman, that means a manager who refuses to soften the message. Arsenal has carefully nurtured him since his debut at 15, and at 16 he is learning the hardest lesson yet.