The Basel striker took a heavy blow to the head late in the first leg of his side's 5-2 aggregate defeat to LASK.
Van Wolfswinkel, who has won two Netherlands caps, was diagnosed with a concussion but sent to hospital for further testing.
Doctors uncovered an aneurysm that otherwise would have gone undetected and the forward will now take time out from football to have treatment.
Revealing the issue to Basel fans in a Twitter video on Thursday, Van Wolfswinkel explained: "Because of the brain aneurysm, I will need treatment.
"Having treatment will mean that I won't be able to play football for a long time.
"At this moment, we are not able to determine a date of returning. It's just clear that it won't be for at least another six months.
"The return date will of course depend on the results and the reaction of the brain aneurysm after the operation. The months after will decide when I could be back."
He added: "I'm just very grateful for the fact that the brain aneurysm got discovered now.
"Normally, a brain aneurysm is not discovered on time, always too late when it has already ruptured or started damage.
"I want to say thank you to the FCB medical department and to the doctors of the university hospital in Basel.
"During the upcoming months, I will keep supporting FCB and my team-mates - like I always have. For now, I will focus on my health and my recovery to do what I love most as soon as possible - and that's to play football."
The 30-year-old, who has previously represented Sporting, Norwich City, Saint-Etienne and Real Betis, as well as Vitesse and Utrecht in his homeland, scored 13 goals in the Swiss Super League last term.