tl;dr A Wilcoxon ranked sum test is not particularly appropriate, but it doesn't really matter because no test will yield a relevant p-value here.
The Wilcoxon test assumes, amongst other things, i.i.d. observations within each group. This is not true of "normalized counts"; a normalized count derived from a large library will be much more precise than the same normalized value derived from a small library. It's hard to predict the effect of this violation, but it's unlikely that you'll get a happy ending here. I'm also guessing that there will be further problems from the excessive number of tied observations at zero.
The "better" approach is to fit a count-based GLM, e.g., with edgeR or friends. You can even do this by subsetting down to the single gene of interest and running that through the pipeline; with a large number of cells, there is no benefit offered by empirical Bayes shrinkage, and so nothing is really lost by ignoring the genes you don't care about. (If you do this with edgeR, make sure you compute the library sizes _before_ subsetting, otherwise your single gene will be used to compute its own normalization factors, thus cancelling itself out. Also, I would suggest using
glmLRT() rather than the
glmQLFTest() recommended for bulk analyses.)
However, all of this minutiae is largely irrelevant because cells are not replicates. Experimental samples are replicates; if someone asked you to repeat the experiment, you'd do it with a fresh set of cells from a new donor/mouse/whatever. Any $p$-value that you get from treating cells as replicates will be much lower than what it should be. In many single-cell contexts, this fact is swept under the carpet because we don't care about the actual value of the $p$-value, we only care about using the $p$-value to rank genes, e.g., for marker detection. In your case, though, it seems you're trying to use the $p$-value to make a statement about significant differences between cell types. Such an interpretation would only make sense if your repeat experiment involved re-sampling from the exact same pool of cells, which doesn't seem as if it would offer general insights into your biological system of interest.
Further comments are provided here. If you do have multiple samples, then you're in a better position, and you can consider reading this.