No, mere competence (or in Voldemort's case, expertise) in spellcasting isn't enough to create the Patronus charm.
JKR noted in a webchat that Snape was the only Death Eater capable of creating a Patronus and only then because he wasn't really in league with them (e.g. due to his love for Lily Potter)
Samantha: Was Snape the only Death Eater who could produce a full
J.K. Rowling: Yes, because a Patronus is used against things
that the Death Eaters generally generate, or fight alongside. They
would not need Patronuses.
In the "Wonderbook : Book of Spells", we see a dark wizard by the name of Raczidian attempt the Patronus and learn the consequences should a "competent but unworthy wizard or witch attempt the spell", namely that you're covered by maggots which engulf your flesh (ouch)
There's been some caveating on Pottermore, regarding the ability of some of Voldemort's hanger's-on (i.e. Umbridge) being able to use a Patronus. In short it's because they're lacking a capacity for introspection...
The Patronus Charm is one of the most ancient of charms and appears in
many accounts of early magic.
In spite of a long association with
those fighting for lofty or noble causes (those able to produce
corporeal Patronuses were often elected to high office within the
Wizengamot and Ministry of Magic), the Patronus is not unknown among
Dark wizards. While there is a widespread and justified belief that a
wizard who is not pure of heart cannot produce a successful Patronus
(the most famous example of the spell backfiring is that of the Dark
wizard Raczidian, who was devoured by maggots), a rare few witches and
wizards of questionable morals have succeeded in producing the Charm
(Dolores Umbridge, for example, is able to conjure a cat Patronus to
protect herself from Dementors).
It may be that a true and confident
belief in the rightness of one's actions can supply the necessary
happiness. However, most such men and women, who become desensitised
to the effects of the Dark creatures with whom they may ally
themselves, regard the Patronus as an unnecessary spell to have in
Voldemort would certainly fall into the latter category. Not only would he consider the spell largely pointless (not to mention unnecessarily risky) but it's unlikely he could summon a 'pure talisman memory' that would satisfy the requirements of the spell given that his soul was so dramatically splintered and corrupted by the various murders he'd committed in the creation of his various Horcruxes.