This article is from the Ozone Depletion: UV Radiation and its Effects FAQ, by Robert Parson firstname.lastname@example.org with numerous contributions by others.
A few physicians have said so, but most others think not.
[Skolnick] [van der Leun and de Gruijl]
First of all, UV-B has not, so far, increased very much, at least
in the US and Europe.
Second, melanoma takes 10-20 years to develop. There hasn't been
enough time for ozone depletion to play a significant role.
Third, the melanoma epidemic has been going on since the 1940's.
Recent increases in rates may just reflect better reporting, or
the popularity of suntans in the '60's and '70's. (This becomes
more likely if UV-A is in fact involved.)