Southampton manager Mauricio Pochettino will make a decision on his long-term future at the end of the season.
He has 17 months left on his deal and said it would make "no sense" for him to leave in the middle of the campaign.
"At the end of the season, we will assess the situation and then we will decide whether I am apt to stay at the football club," he told BBC Sport.
"I am fully committed until the end of the season and taking this club as high as possible.
"It is only normal that we finish the season and then we will see what happens."
On Thursday, the Argentine met club owner Katharina Liebherr, who installed herself as non-executive chairman following Wednesday's departure of Cortese.
Pochettino sought assurances about the club's future direction amid rumours Liebherr is keen to sell.
It was only his second meeting with her since he replaced Nigel Adkins as manager a year ago.
The meeting lasted only five minutes, but Liebherr insisted she fully supported the 41-year-old manager, who admitted to a sleepless night following the Italian's exit.
Pochettino said in the summer he would leave St Mary's if Cortese departed, but explained the circumstances were very different now.
And later in the news conference he said he could still see out his contract at the club.
"They are different moments," he said. "When I came here, it was to keep the club in the Premier League. After that, a new project started.
"It would make no sense for me to leave the club in the middle of the season. No-one would understand it and I'm sure Nicola would not understand it.
"I have a commitment to the club, the staff, the supporters. My contract is with Southampton and I intend to see out my contract with this company. It would not be the right moment to leave this club right now.
"When people leave, things change, but my responsibility to this club is to make sure it is always up. I am a loyal person and I am a legal person. I follow law."
Pochettino said he would fight to keep his star players, including defender Luke Shaw, at St Mary's during the January transfer window.
"No-one is for sale at Southampton," he said. "No player that I don't want to leave this club, will leave."
But he admitted Cortese's departure was a blow, both personally and to the club, and had made him consider his short-term future at St Mary's.
"Of course another person is going to be appointed and they will be different to Nicola," added Pochettino, who said he had spoken to Cortese about his future.
"It's clear Nicola and I had a very good relationship. At night, I was thinking about many things, including my position at the club. It was a heavy blow to digest."
Southampton legend Matt Le Tissier welcomed Pochettino's move to reassure fans and criticised the former chairman for "not treating people like human beings".
"The most important thing at the football club are the players and manager, they are the ones that get results," said Le Tissier, who represented the club for 16 years until 2002.
"As long as their focus has not been diverted away from the job in hand then I don't think there is too much to worry about.
"The biggest worry for me is if the Liebherr family decided to sell the club, that would put a whole different perspective on things, but as it stands at the moment we've lost an executive chairman and I don't think it's quite as big a deal as Nicola's PR team are making out."