There are a couple problems with the question that become apparent when we break it down into one thing that is implied, and another that is the object of the question.
The implication is that there is global warming to begin with, and along with that, how bad it is if it does exist. Further, there is the side question of just what caused it, again assuming that it exists. Let's look at that first.
There is some evidence, though not well founded, that the global temperatures have risen some tenths of a degree, Fahrenheit, in the past century. This is a point of science fact, though it is also science fact that the temperature isn't raising fast nor a great deal. Frankly, we are still guessing about the cycles on Earth, on the sun, and having to do with our global temperature. We just don't know if this is a normal fluctuation. Part of that confusion is that the same sources of data show that Earth has been far warmer, globally, in the past than it is now. In fact, there is good evidence that the global temperature was several degrees warmer two thousand years ago than it is today.
If we define global warming as any increase in global temperatures, no matter how slight and regardless of the cause, we'd have to conclude that there is global warming. However, if we define it as an increase in temperature that goes beyond normal fluctuations, we'd have to conclude that there is no global warming, based on our knowledge at this time and the amount of data that we have.
Regardless of the definition, proponents of the global warming theory like to point out that glaciers and ice caps are melting. This part is very true. It is measurable. Still, this is not proof of global warming. The reason is often totally ignored. Ice packs and glaciers are produced by the accumulation of snow. If snow doesn't accumulate, or the speed at which it does, becomes less, the ice packs will dwindle from the normal melting that goes on every year. Like anyplace on the earth, the polar caps can suffer drought, and in fact, Antarctica qualifies as a desert, since it receives only a few inches of precipitation a year.
Look at this data a little more deeply, and you will see that glaciers and ice pack have been on the decline, gradually, since at least the 18th century. There was a brief buildup during the little ice age, but the rest of the time, ice packs have lessened. Science has shown that even this isn't constant, but that it isn't happening appreciably faster now than it did a couple centuries ago, before cars and other things that are commonly blamed by the global warming proponents even existed.
So now that we've seen how much confusion even surrounds the concept of global warming, lets look at the connection with the ozone layer. Simply put, it is very difficult to see any correlation between the two. Put in a slightly different way, weakening of the ozone layer could encourage global warming, since it would allow more ultraviolet radiation to reach the surface, it doesn't work the other way around.
Oxygen rarely exists as single atom. It combines readily with other elements, including itself, to make an oxygen molecule, O2. If an O2 molecule gets enough energy, it can combine with oxygen atoms to form O3, or ozone. The distinctive smell that happens during electrical arcs is because of the formation and presence of ozone, and the electricity furnishes the energy to make this possible.
Ozone is unstable. Even sunlight breaks it down into oxygen molecules and single oxygen atoms. This occurs yearly in the ozone layer. Heat is a form of energy, but rather than destroying ozone, it creates it. Still, the amount of temperature increase necessary for this to happen is far greater than anything this world has ever witnessed on a global scale.
People tend to over react and overstate things. Hence, there is talk about the ozone "hole" over the south pole. We now have plenty of scientific data to show that not only is this a normal yearly fluctuation, but it isn't a hole at all, it is merely a thinning of the ozone that occurs every year. Keep in mind that the south pole, and the north pole for that matter, go for 6 months each year where they get nothing even approaching direct sunlight, and 6 months where the sunlight is almost constant. It only stands to reason that for 6 months, the ozone layer in those locations weakens, and for the next 6 months, it strengthens. This is exactly what scientists have found.
Man indeed has had an impact on the ozone layer, but not like people think. CFC's or chloro-floro-carbons, do destroy ozone. Generally, though, they don't raise high enough into the atmosphere to cause damage. But man is not the only producer of CFC's. Volcanoes also produce them, and they have the energy to get them into the ozone layer, where they cause damage. CFC's interact with the free oxygen atoms, preventing them from being available to produce ozone. But volcanoes are a natural process that has been going on for billions of years.
There really isn't any connection between global temperatures and the "health" of the ozone layer. During the time of the dinosaurs, the mean global temperatures were some 5 to 10 degrees higher than they are now, yet there was a hefty ozone layer at the time, otherwise the dinosaurs could not have survived.
Whether a person chooses to believe in global warming or not, it is pretty easy to see, using science, that global warming or cooling has nothing to do with the ozone layer.