Can water be ignited in an ozone atmosphere?
We are basically talking about a derivation of the $\ce{H2/O2}$ system, which involves a radical mechanism. You maybe want to have a look at this answer of mine going a bit more into detail.

The reaction you are interested about is only a total sum of reactions and is most likely not to occur as other reactions will be favourable. Ozone itself is not a very stable compound and many other molecules may actively catalyse decomposition. There are various publications[1-3] on the mechanism of this reaction:
\begin{align}\ce{
  M + O3 &amp;&lt;=&gt; M + O2 + O\\
  O + O3 &amp;&lt;=&gt; 2O2\tag1
}\end{align}

The range of $\ce{M}$ is in $\ce{O3, O2, CO2, N2, He, etc}$.[1] Although not stated explicitly, there should basically no reason why water is not be able to accomplish the same. Hence the driving force of the total reaction would be entropy, giving the overall equation:
$$\ce{2 O3 -&gt;[\ce{H2O}] 3 O2}\tag2$$

It is most likely that the reaction will start with the formation of oxygen radicals in the gas phase ($\ce{O}$ atom transfer).$$\ce{O3 + O3 -&gt; O3 + O2 + O}\tag3$$

In presence of Hydrogen donors ($\ce{RH}$ apparently any kind will do) a chain reaction will be initiated.[1] $$\ce{O + RH -&gt; OH + R}\tag4$$

The main chain carriers will be $\ce{HO, HO2}$.
\begin{align}\ce{
  HO + O3 &amp;-&gt; HO2 + O2\\
  HO2 + O3 &amp;-&gt; HO + 2O2\tag5
}\end{align}

The process will be terminated by this (quite unspecific) reaction. This may then also involve $\ce{H2O2}$, which "is expected to be an excellent $\ce{H}$ donor."[1]
$$\ce{2HO2 -&gt;}\text{stable products}\tag6$$

In aqueous solution the decomposition is highly pH dependent, but may always work because of autoprotolysis. $$\ce{2H2O &lt;=&gt; H3+O + {}^{-}OH}\tag7$$

Steahelin and Hoigné propose several hypothesis in which way ozone will be decomposed, but I will not go into absurd detail.[4] The core assumptions involve $\ce{{}^{-}OH}$ as initiator of a radical chain.
\begin{align}\ce{
  O3 + {}^{-}OH &amp;-&gt; .O2- + HO2. &amp;  HO2. &amp;&lt;=&gt; H+ + .O2-\\
  O3 + {}^{-}OH &amp;-&gt; HO2- + O2   &amp;  H2O2 &amp;&lt;=&gt; HO2- + H+\\
                               &amp;&amp;  O3 + HO2- &amp;-&gt; .OH + .O2- + O2\\
  O3 + {}^{-}OH &amp;-&gt; .O3- + .OH\tag8
}\end{align}

Which is basically followed by another set of radical chain reactions.
\begin{align}\ce{
  O3 + .O2- &amp;-&gt; .O3- + O2\\
  .O3- + H2O &amp;-&gt; .OH + {}^{-}OH + O2\tag9
}\end{align}

Which is all again resulting in $(2)$ again.

In summary the proposed reaction might occur, but will immediately be followed by another set of reactions resulting in the formation of biatomic oxygen.

References


Sidney W. Benson, and Arthur E. Axworthy Jr., J. Chem. Phys., 1957, 26, 1718.
J. V. Michael, J. Chem. Phys., 1971, 54, 4455.
Oliver R. Wulf, and Richard C. Tolman, *Proc. Natl. Acad. Sci. USA, 1927, 13 (5), 272–275.
Johannes. Staehelin, and Juerg. Hoigne, Environ. Sci. Technol., 1982, 16 (10), 676–681.