Victor Osimhen nodded home to give Napoli the lead in the 51st minute before Udinese forward Isaac Success bundled in the leveller in stoppage time.
"Clearly I thought we could do better, without a doubt. Unfortunately, that wasn't the case," coach Francesco Calzona told DAZN.
"We have to ask the management about the future of Napoli, it's clear that we have the qualities to do better. It was a disappointing and unfortunate year, but we absolutely must finish this championship on a high note."
This may be challenging as Napoli sit eighth on 51 points after 35 games, with nine points separating them from both fifth-placed Atalanta and AS Roma, in sixth.
They had earned 50 points halfway through their winning 2022/23 campaign.
"Naples is a great place, perhaps after a year like this there will be many rumours circulating," Calzona said.
"But it is not a justification, I repeated that there is still a possibility of reaching Europe.
"We have improved a little, but it's not enough. We have to do more to win these games. We have also created, the team continues to have good possession, at times sterile.
"But we have to be more incisive. In the second half we did it but we have to avoid the end of the game where it gets dirty".
Napoli next host fourth-placed Bologna on Saturday before they travel to Fiorentina, in ninth, who have a game in hand.
