Are gem-diols on alkenes stable?
What you call a "gem-diol alkene" might also be called an "ene-diol" or "endiol" by analogy to the name enol.

Enols are a tautomeric form of aldehydes or ketones.  They have one double bond and one C-O single bond.  Since "ene-diols" have one double bond and two C-O bonds, the oxidation number of the O-bonded carbon is 2 higher than with enols.  This corresponds to the oxidation number of carboxylic acids.  Thus, "ene-diols" are (theoretically) tautomers of carboxylic acids.

Ene-diols are not stable because they would rapidly tautomerize the more stable carboxylic acid form. 

Hydrogen bonds like the one you have drawn are unlikely to stabilize enols very much, for reasons that Jan noted in a related question here on chem.SE.