Yes, he is the one that was pushing all along for this to be taken off the list of planets, but he was not wrong in his ideas. Pluto is not a real planet, and he seen this and he was the one to let the whole world know it.
I do not believe you can single out Neil DeGrasse Tyson as the person responsible for the demotion of Pluto. Did he support the classification change? Yes, he did. With that said, I think it is important to understand that this change in thinking occurred because there have been several other objects, just like Pluto, found, one that is even larger than Pluto. I believe it was better to change it's classification so it can be correctly grouped with other similar objects, rather than adding a whole bunch of small planets to the line-up that barely fit the description.
No, I do not believe that Neil DeGrasse Tyson is, in fact, responsible for the demotion of Pluto from planetary status, because there were many people that were involved in the decision to demote Pluto. Tyson does not work in a vacuum. There are many scientists that Tyson made the decision with, based on changing criteria.
I believe the debate about whether or not Pluto was a proper planet had been going on for many years before Neil Degrasse Tyson made any remarks, I believe sometimes with enough publicity or high profile figures making comments that it may push the processes along but I don't believe he is responsible for Pluto's demotion.
I think that while Neil DeGrasse Tyson has some part in the demotion of Pluto from planetary status, I believe it happened moreso due the scientific community agreeing that Pluto should not technically be considered a planet. Neil DeGrasse, however, could be considered the face for the initiative that brought upon the situation.