Scotland ran in seven tries to sweep aside their Eastern European opponents with ease in the first of their November Tests.
But the gloss will be taken off the win if White, who set up the first try for Johnnie Beattie, is ruled out of the rest of the series through injury.
With the muscular and hard-hitting Pacific Islanders up next for Hadden's men, the absence of the abrasive Sale flanker - an inspiration in Scotland's revival over the past year - would be a considerable blow.
Hadden said: "Jason has had an initial assessment and he will be getting a scan on Monday morning so we really won't know until then.
"The injury happened when he got his studs caught in the turf leaving a scrum and he twisted awkwardly. There was no-one involved in a contact situation.
"He's feeling a bit foolish and a bit sore and we're keeping our fingers crossed that it is not serious. It was obviously serious enough for him to come off and it was difficult for him to bear any weight on it.
"It doesn't look a good sign but we'll just have to wait and see and we won't know any more until Monday."
Hadden was pleased with the performance as a whole and paid particular tribute to new caps Beattie and Rob Dewey - both of whom scored debut tries.
With Phil Godman and Euan Murray also handed their first starts the comfortable victory was achieved with several unfamiliar faces in the side.
Hadden said: "I thought it was a good performance against very difficult opposition. Although we have managed to score seven tries they made us work very hard for it.
"The quality of some of the tries was very pleasing and although there will be harder tests ahead, I thought the intensity and the physicality of the game was like a real international.
"I was pleased that the opposition was as strong as they were because then you get a better idea of how people are doing."