The Colombian, on a season-long loan at Napoli, sustained a blow to the head when he collided with Ignacio Pussetto early in the match.
He then fell to the ground shortly before the break after Udinese levelled at 2-2 having been 2-0 down.
The 30-year-old received treatment before he was brought off the field.
Napoli, who beat Udinese 4-2, later released a statement revealing that Ospina would remain in hospital overnight under observation.
Coach Carlo Ancelotti said: “Fortunately it’s nothing serious, he’s conscious. They did a CAT scan that is negative.”
Ospina signed for the Gunners in July 2014 but has only made 27 league appearances for the Premier League side.
He joined Napoli on a season-long loan deal in August 2018.