The view that the fish that swallowed Yunus (peace be upon him) is still alive because Allah, may He be Exalted, says: {And had he not been of those who glorify Allah, he would have remained inside its belly until the Day they are resurrected} [As-Saffat 37:143-144] is not a sound view, because it is a view for which there is no evidence, and we do not know of anyone among the scholars of tafsir (commentary on the Qur’an) and Arabic language who said that.
Quoting these verses to prove that is not correct, because if the stipulation for something to not exist is fulfilled, then the thing that could have existed will not materialise.
For more information, please see Al-Itqan fi `Ulum al-Qur’an, by as-Suyuti, 2/284.
Because Yunus (peace be upon him) used to glorify his Lord a great deal, it was impossible that he could have stayed in the belly of the fish until the Day they are resurrected. This verse is similar to the verse in which Allah, may He be Exalted, says: {Were it not for a prior decree from Allah, a grievous punishment would have befallen you for what you took} [al-Anfal 8:68]. The punishment did not happen because of the prior decree from Allah.
And Allah, may He be Exalted, says: {And if We had not strengthened you, you would have almost inclined to them a little} [Al-Isra’ 17:74].
The Prophet (blessings and peace of Allah be upon him) could not have inclined towards the polytheists because Allah had strengthened him.
The conclusion of all of this is:
Yunus (peace be upon him) was one of those who glorify Allah a great deal, and because of that Allah saved him from those layers of darkness, and he did not remain in the belly of the fish for the length of time mentioned, which could have happened if he had not been one of those who glorify Allah a great deal.
Moreover, there is no connection between any of that, directly or indirectly, with the idea that the fish could still be alive today, or the question of when it died, or whether it died or not. All of that is a matter of conjecture that never materialised.
Furthermore, it could be said that staying in the belly of the fish until the Day they are resurrected does not necessarily mean that the fish could have continued to live until the Day of Resurrection. It is not impossible that the fish could have died and the human in the belly of the fish could have died, then he would remain in its belly until the Day of Resurrection as the deceased remains in his grave until the Day of Resurrection.
And Allah knows best.