Thomas More looks at oaths with a level of intensity that we usually reserve for late-night sessions of Grand Theft Auto Online. That is to say: he takes them really, really, really seriously.
Just listen to the man himself:
MORE: [...] When a man makes an oath, Meg, he's holding how own self in his own hands. Like water. (He cups his hands) And if he opens his fingers then—he needn't hope to find himself again. (2.607)
Serious stuff. To the totally law-abiding Sir Thomas More, an oath is a direct reflection of one's inner self. What's more, a failure to respond honestly to oaths could lead to an individual not even knowing what he or she believes anymore, which is a scary thought. Do that, and you'll end up like Norfolk and his aristocratic bros, shifting your beliefs whichever way the wind blows. (And things don't turn out too well for Norfolk in the end, as we find out.)
More, on the other hand, demands to know the exact wording of an oath before he even considers signing on the dotted line. This exchange with Roper is tells us pretty much everything we need to know:
ROPER: We don't need to know the [...] wording—we know what it will mean!
MORE: It will mean what the words say! An oath is made of words! It may be possible to take it. Or avoid it. (2.451-452)
While More has no issue signing an oath of allegiance to King Henry in theory, he'll refuse if it contains even one assertion he disagrees with. That's why he ultimately doesn't sign it: he disagrees with its claim that King Henry—a political figure—has power over religious matters. Whether or not you agree with this argument, More's refusal to deny his beliefs in the face of death shows how seriously he treats oaths.