"The morality of it" is a noun phrase; "is debatable" is a verb phrase.''
You cannot contract words across phrases.
Right: "You do what you can, not what you should."
Wrong: "You do what you can't what you should."
Edit: Greg Lee makes the pedantic but surprisingly productive point that individual nouns are technically phrases. I was just going to concede his point and re-write my rule as "Do not contract words across multi-word phrases", but then I thought, is that really the rule?
My example above is a contraction across clauses, which is clearly not sensible, but consider the following sentence in conversational English:
"Ice cream'd be delicious right now."
Contracting the "would" into the noun makes perfect sense there. Now consider
"A scoop of ice cream'd be delicious right now."
Is that really less sensible? The noun phrase is quite complex, a prepositional phrase tacked onto a noun and an article. The "would" should contract with "scoop", but since that isn't structurally possible, we tack it onto whatever word is physically at the end of the phrase.
The difference between "a scoop of ice cream'd" and "the morality of it's" is that "it's" is also a pre-existing contraction, just not the one we want, so the latter is more distracting; it "feels" wrong.
In conclusion: jeez, I dunno. Write "its morality is debatable" and you don't have to worry about it.