Essentially More is an Honest man, as King Henry VIII points out to him in their conversation early in Act 1. That need to remain honest extended from a fear of going to hell. This was the 16th century and everyone, even the King himself fully believed that they would be judged by God for their actions. Henry got himself in a lather over the reading from Leviticus that said Thou shalt not uncover the nakedness of thy brother's wife. He wanted a son as the throne was not believed secure under a woman at this time.
People also believed that bad luck was a sign of God's disatisfaction with you. Henry took the absence of a living son (Sons were wither stillborn or died in infancy) as Punishment for imploring the Pope to make an exception to allow him to marry Catherine in the first place.
However More believed that the King was wrong, and that to make the Pope oveturn his own ruling was primarily afronting to the Pope. Then to deny the absolute supremacy of the Pope as God's voice on earth would be to deny God himself, thereby dooming More to Damnation.
More is motivated by integrity. He will not bow to those in power when he believes the leadership to be less than laudable.
More feels that the Pope is corrupt and to comply with his edicts is morally unjustifiable. More explains, "What matters is not that it's true, but that I believe it; or no, not that I believe it, but that I believe it." More fears that if he breaks with his conscience, he will be damned to hell, while his associates and friends are more concerned with holding onto their own temporal power.
As for the specific objections to the Acts of Supremecy, More argues with Norfolk, his good friend who has bowed to the pressure of signing the Acts which he does not truly believe in:
Norfolk: Look, I'm not a scholar, and frankly I don't know whether the marriage was lawful or not — but Thomas, look at these names! You know these men! Can't you do as I did and come along with us for fellowship?
More: And when we stand before God, and you are sent to Heaven for doing according to your conscience, and I am sent to hell for not doing according to mine, will you come along with me — for fellowship?"
Thomas More is a "man of selfhood". He was much sainted and known for his refusal to swear an oath to King Henry's supremacy to the Pope. He is the main protagonist of the story, and also the hero of it. He looks into his inner self and being for motivations and support without the external influence of other people's ideals to influence his decisions and actions. If an idea or thought is good to his conscience, he would follow and live up to it, but if it's the opposite, he would not even take a second look and discard it away. If the person in power has poor leadership qualities and not fit to rule over the country, he would not submit to his power and control. This shows that he has his own moral core values and he won't get pushed around by other people, only with himself