'morally problematic' is correct and usable in written English.
You can use it when referring to a situation that presents ethical or moral dilemmas. For example: "The situation of Bob revealing the secret about Jane to her parents is morally problematic."
However, there is a set of separable questions about personal morality: about (for example) whether buying an organ for oneself might be morally problematic even if it should not be prohibited.
Since you consent to this exchange, there's nothing morally problematic about it.
The chapter entitled "The Making of a Youth Culture" is morally problematic, as is the period it describes.
The lesser-evil position, Ignatieff tells us, lies in never losing sight of the "morally problematic character of necessary measures".
I agree that abortions are morally problematic, but that does not justify the complete disregard for women's rights.
And that is not only because it is morally problematic for the United States to reach out to the Iranian regime under the current circumstances.
"The relevant question," Professor Levinson wrote, "is whether collective sanctions are morally problematic in a way that individual ones are not".
To focus on the milk and honey rather than the blood and guts is a willful refusal to grapple with the morally problematic nature of the Bible.
"I Love You Dude" is thought-provoking, extremely wordy and morally problematic to those of us who are nitpicky about such things.
Half of US Mormons are also different from vast numbers of Americans in not drinking coffee and regarding alcohol as "morally problematic".
It is always morally problematic when society's most vulnerable members, children, have their images taken from them without being able to refuse or consent.
