EDIT--Hmm, well I've decided to edit this, so as not to offend anyone. Suffice to say, both sides seem able to bring forth statements of St. Gregory to evidence their position... so...
I haven't seen any evidence here to suggest that Gregory believed in Roman supremacy.
I gave a link that provided quotes such as:
"Inasmuch as it is manifest that the Apostolic See, is, by the ordering of God, set over all Churches, there is, among our manifold cares, especial demand for our attention..."
"To all who know the Gospel it is clear that by the words of our Lord the care of the whole Church was committed to Blessed Peter, the Prince of the Apostles... Behold, he received the keys of the kingdom of heaven, the power to bind and loose was given to him, and the care and principality of the entire church was committed to him..."
I can direct you to a watering hole, but I suppose I can't make you drink...
If you do a Google search I'm sure you could many more pages on the topic. But you'll have to find your own way to the next watering hole