The reaction that is not involved in the ozone layer depletion mechanism in the stratosphere is:

First, we recall what the ozone-depletion mechanism looks like in the stratosphere. The chain is initiated by highly reactive free radicals, most famously the chlorine radical $$Cl^{\bullet}$$ that comes from chlorofluorocarbons (CFCs). The key elementary steps are:

$$\text{(1) } Cl^{\bullet}(g) + O_3(g) \rightarrow ClO^{\bullet}(g) + O_2(g)$$

$$\text{(2) } ClO^{\bullet}(g) + O(g) \rightarrow Cl^{\bullet}(g) + O_2(g)$$

Adding the two reactions, the $$Cl^{\bullet}$$ cancels, giving the net destruction of ozone:

$$O_3(g) + O(g) \rightarrow 2O_2(g)$$

The radical is thus regenerated and can repeat the cycle many times, so even a trace amount of $$Cl^{\bullet}$$ can destroy a large amount of ozone. Because the radical must come from somewhere, we also need a photochemical source reaction, for example

$$\text{(3) } CF_2Cl_2(g) \xrightarrow{uv} Cl^{\bullet}(g) + CF_2Cl(g)$$

or

$$\text{(4) } HOCl(g) \xrightarrow{h\nu} OH^{\bullet}(g) + Cl^{\bullet}(g)$$

Both reactions (3) and (4) furnish the chain-propagating $$Cl^{\bullet}$$, so they are definitely parts of the overall ozone-depletion scheme.

Now, let us see whether each of the four options given in the problem participates in this mechanism.

Option B is reaction (2) exactly, so it is clearly on the list.

Option C is reaction (3), the photodissociation of a CFC, again essential.

Option D is reaction (4), another photolytic source of $$Cl^{\bullet}$$, hence also relevant.

Option A, however, is

$$CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O$$

This is the ordinary combustion (oxidation) of methane. It involves no chlorine-containing species and generates no $$Cl^{\bullet}$$ or $$ClO^{\bullet}$$ radicals. Therefore it does not feed into the catalytic destruction cycle of ozone in the stratosphere. Combustion of methane takes place mainly in the troposphere and is unrelated to the specific free-radical chain we have just discussed.

So, among the four reactions, only Option A is not involved in the ozone-layer-depletion mechanism.

Hence, the correct answer is Option A.