The Potters agreed a deal for the US international winger last week, but he has not played since August following foot surgery.
Stoke were keen to check on the fitness of the 22-yaer-old before finalising the deal, but Major League Soccer was fearful that Shea would not stand up to a rigorous medical.
Pulis said: "We've not managed to get Brek Shea over here. That's over. We'll draw a line under it now."
Another possible arrival that could be scuppered due to injury is Dutch international Erik Pieters, who will be out of action for around six weeks with an injured arm.