Generally, no

There is no ruling of any kind in any official 5E material as far as I'm aware.

With what is given in the description of Petrifying Gaze, I'd rule that the head must be preserved and/or magically prepared in a way that would make it more than just a slowly rotting head.

Petrifying Gaze. When a creature that can see the medusa's eyes starts its turn within 30 feet of the medusa, the medusa can force it to make a DC 14 Constitution saving throw if the medusa isn't incapacitated and can see the creature.

Emphasis mine

Personally, I'd interpret 'not being incapacitated' as a state of being that the head of a dead medusa can't retain on its own.

Regarding her ability to petrify I'd consider it a byproduct of her curse that manifests whilst intently staring at a creature. Regarding arcane preservation/preparation I see no reason why a player shouldn't be able to have the head focus its gaze on a location/creature the wielder wills it to.

To prevent the petrification from being too powerful a tool you could limit the amounts of times the head can be made to focus, lower the DC or even prepend the DC with a mock ranged attack, as focussing someone else's eyes manually upon another creature's eyes.

As suggested in the comments by @Pixelmaster I'll add a brief chunk regarding the itemization of the head. I'll be referring to the

Flesh To Stone, a 6th level spell, would be the closest equivalent to Petrifying Gaze for players. The closest RAW item dealing a comparable status would be the rare Wand of Paralysis (Dmg P211), which has 7 charges and paralyzes a single Creature for one minute. If you wanted to you could base the stats of the head on the wand like so:

Medusian Stare

Preserved Head, Rare

This head has (2-5) charges. While holding it, you can use an action to expend (1/2) of its charges to cause the head to briefly animate and stare deeply into the eyes of a creature of your choosing, forcing it to pass a DC (12-14) Con save. If the saving throw fails by (5-7) or more, the creature is instantly petrified.

Otherwise, a creature that fails the save begins to turn to stone and is restrained. The restrained creature must repeat the saving throw at the end of its next turn, becoming petrified on a failure or ending the effect on a success. The petrification lasts until the creature is freed by the greater restoration spell or other magic.

Alternatively you can expend (3/4) charges to have the head target all creatures in a (15-20) foot cone in a direction you point the head.

If all charges are expended or the Medusa's head sees itself reflected on a polished surface within (20-30) feet of it and in an area of bright light, roll a d20. On a (1-4) the medusa's head is, due to its curse, affected by its own gaze, destroying itself in the process.

The head regains 1d4 expended charges daily at dawn.

Note that the number of charges and their expenditure are deliberately vague. While a Wand of Flesh to Stone would be a rare item like the Wand of Paralysis their respective Spell versions are quite far apart. As far as I see the Wand of Paralysis is based on Hold Person, a 2nd Level spell, only the want targets Con instead of Wis. Giving a higher level spell less, or maybe even just one use per day may be the way to go.

Another thing to keep in mind would be that preparing the head for use should be a journey of its own, a little sidequest to discover some lost knowledge the character needs to force a spark of life back into the eyes of the medusa. You could also have the character roll several times through the creation of the item and have the Charges and DC based on how well they do.