It was a bombshell reveal when Aang unmasked Zuko as the Blue Spirit in Avatar: The Last Airbender, and despite the Fire Nation prince being his enemy at the time, Aang never told anyone Zuko's secret. Being the Avatar, Aang was public enemy number one according to the Fire Nation. They had been searching for him for a hundred years, and Zuko was tasked with capturing and delivering Aang to his father in order to be accepted back into his good graces. That made them enemies.
But despite Zuko's charge, he was never fully committed to the idea of taking down the Avatar. This was something he struggled with for the entirety of the series, before he ultimately made his choice and sided with Team Avatar. Even early on, though, he exhibited some hesitation. When Commander Zhao captured Aang and held him at Pohuai Stronghold, Zuko broke into the compound disguised as the Blue Spirit and freed Aang. However, an explosion went off near them during their escape, which knocked Zuko unconscious - and Aang took this opportunity to unmask his savior, revealing Zuko was the Blue Spirit.
When Aang returned to Katara and Sokka later on, he refrained from telling them what he found out. Instead, Aang kept Zuko's secret for the rest of the series, which was proven by the Ember Island Players' play, The Boy in the Iceberg. Much of the Ember Island Players' breakdown of the story was accurate, albeit heavily dramatized, but they were mistaken about the fight at Pohuai Stronghold. According to the play, Zuko captured Aang, who was later freed by the Blue Spirit. They got the latter part right, but it was Zhao who captured him. The Ember Island Players didn't know that Zuko was the Blue Spirit, which in turn means no one found out about what really happened that night.
Aang never harbored any animosity towards Zuko even when the prince was chasing after him; a huge part of that delves from Aang's upbringing with the Air Nomads. He developed an anti-violence personality and chose to welcome everyone, despite their position on the Avatar and their way of life. This was especially evident by Aang finding an alternate way to defeat Fire Lord Ozai, rather than take his life (which most people in the world would've found justified).
All of this explains why Aang never gave up Zuko's secret. The Blue Spirit was a force for good; Zuko never used firebending while in costume, instead choosing to wield two swords (after being taught who to fight with swords by Master Piandao). On some level, Aang knew this and he knew giving up Zuko as the Blue Spirit could have disastrous consequences for him. Of course, a handful of people did find out that Zuko was the Blue Spirit, but none of them found out from Aang.