Mike Mearls tweeted that it should be a minimum of 1 point, but I can't find the same from Jermey Crawford. Mike Mearls talked about what was intended, but it isn't an official ruling:
no, should be minimum 1
Jeremy did weigh in about hit dice though:
You regain no hit points if you spend a Hit Die and your Constitution modifier reduces the total to 0 or lower. #DnD
If Mike is right about this spell, then the player becomes stable. If Jeremy's tweet about Hit Dice also apply to spells like cure wounds there would be no reason to assume that healing for 0 would stabilize someone, as the spell text doesn't say anything of the sort; but I think as a DM I'd let it.
Updated Text of Hit Dice and RAW
When the question was asked Jeremy's tweets were considered official rulings. Since then the text of regaining health from hit dice was edited in later editions since the question was asked to include:
The character regains hit points equal to the total (minimum of 0).
The text of the spell was not updated. So the only clues we have about cure wounds is the word "regain" in the spell, which implies it cant' be negative, and the intent text above. Unless a word has a special game meaning, the common definition applied. Oxford Languages Dictionary offers the following definition:
re·gain /rəˈɡān/ (verb)
obtain possession or use of (something) again after losing it.
"she died without regaining consciousness"
You can't obtain possession or use of negative hit points. Any argument must also use that common meaning standard. So when the rules say things like:
When a creature receives healing of any kind, hit points regained are added to its current hit points
It is telling you what to do, add them to your total, after you regain them. This doesn't change the meaning, define or redefine the word regain.