The 28-year-old was taken to hospital for tests after being briefly knocked unconscious during the first half of the World Cup qualifier in Baranquilla.
Martinez hit his head on the pitch after claiming the ball in an aerial challenge with Everton centre-back Yerry Mina.
Argentina were 2-0 up at the time but after Martinez was taken off on a stretcher ended up drawing 2-2, conceding a stoppage time equaliser.
The goalkeeper, who was allowed to return with the rest of the squad to Buenos Aires after being checked over, said: “I’m better. It was a strong hit but the doctors did their job.
“Obviously I wanted to continue playing, the anger of being taken out, and I found out that they drew at the end. It’s a bitter taste but I’m fine.
“They took me out because I lost consciousness a little; I hit the back of my neck hard but the crying and the anger was because it was my second match with the national team and I didn’t want to be substituted. But the doctors did their job.”
Martinez was making just his second appearance for Argentina, having made his senior international bow in last weekend’s 1-1 draw with Chile.
He will now resume preparations for their Copa America campaign, which begins against Chile on Monday.
Martinez, who joined Villa in a £20million move from Arsenal last summer, is not the club’s only representative at the tournament.
Midfielder Douglas Luiz will be aiming for glory with Brazil and the 23-year-old was a second half substitute as they defeated Paraguay 2-0 on Tuesday to remain top of the World Cup qualifying group.
Brazil kick-off the Copa America when they face Venezuela on Sunday night.