Is what the questioner did—considering what came out of him during sleep as urine, which necessitates ablution (wudu') and not ritual bathing (ghusl)—a correct action?

If a sleeping person sees semen, ghusl (definition) (ritual bathing) becomes obligatory for him, even if he does not recall having a wet dream or experiencing pleasure. The Shafi'is obligate ghusl simply upon the emission of semen. If you are uncertain about the discharge—for instance, if it could be semen, wadi (thick white discharge), or madhi (pre-seminal fluid)—then you have the choice between their respective rulings: you may perform ghusl or perform wudu (ablution) and wash what it has touched.