Probably because <PERSON> considered <PERSON>'s Ring useless to him. <PERSON> says of the Three Rings generally,
[T]hey were not made as weapons of war or conquest: that is not their power. Those who made them did not desire strength or domination ... but understanding, making, and healing, to preserve all things unstained.
from "The Council of Elrond" in "The Fellowship of the Ring"
At the time he gave it to him, <PERSON> tells <PERSON> about <PERSON> specifically,
For this is the Ring of Fire, and with it you may rekindle hearts in a world that grows chill.
from "The Tale of the Years", Appendix B to "The Return of the King"
<PERSON>'s words suggest that Narya's most important power (or at least its power that would be most important to <PERSON>) is an ability to awaken hope and the will to survive in the hopeless. No doubt its power was at work when <PERSON> and the knights of Dol Amroth rallied the spirits of despairing Gondorians during the siege of Minas Tirith.
<PERSON>, as Middle-Earth's second greatest authority on the Rings of Power, would have known all this; and known, too, that the Three Rings of the Elves "endure no evil", as <PERSON> says to <PERSON>. Perhaps this last consideration carried weight with him. 381
His words to <PERSON>
We can bide our time, ... deploring maybe evils done by the way, but approving the high and ultimate purpose: Knowledge, Rule, Order
from "The Council of Elrond"
show that he is aware that he is doing evil, whether or not he still deceives himself that good is his ultimate aim.
<PERSON> characterizes <PERSON> thus:
He is plotting to become a Power. He has a mind of metal and wheels; and he does not care for growing things, except as far as they serve him for the moment.
from "Treebeard" in "The Two Towers"
<PERSON>'s attempt to desolate The Shire seems to bear out <PERSON>'s description. Making, healing, and understanding are not high on his priority list.
In any case, at the time <PERSON> captured <PERSON> he hoped for either of 2 outcomes: if <PERSON> triumphed, to use the power of his Voice to become the power behind <PERSON>'s throne; or to use the One Ring to crush <PERSON> and become ruler of Middle Earth in his own right. <PERSON> knows that even with the Ring of Fire, <PERSON> is no match for <PERSON>; he probably won't be either, so it can't help him defeat <PERSON>. Nor was it likely to help him manipulate <PERSON>; and in the event of <PERSON> becoming <PERSON>'s servant, there would be a very real danger of <PERSON> discovering that his new adviser possessed one of the Three Rings -- with probable bad consequences for <PERSON> and his plans. 381