obviously, western blot is semi- quantity technique that used to identify proteins. In my opinion I don't see any wrong use this technique to detect your protein of interest as long as you have good AB and I would optimise the concentration of AB and I would think carefully the concentration of the gel that you suppose to use.
I looked around a bit and I have the impression that western blot for immunoglobulins is not commonly done, that's why I was wondering whether it maybe wouldn't work well for some reason.
this paper might give you some idea (http://www.fasebj.org/content/21/11/2931.full.pdf), the antibodies do exist. You don't say which cell lines you are talking about, but IgG's etc are mostly excreted/ in the serum eg. I have isolated an IgG-related tagged protein from cell culture media [after being excreted from the cells, post-transfection] using a protein A column. They are often glycosylated too, so it looks like a smear on the blot sometimes.
You would need to consider whether you want reducing or non-reducing conditions (ie use of DTT, do you want to probe for the heavy vs light chains? For proper reducing conditions use 50mM DTT in your sample loading buffer).
As I said above, you may be able to pull down your IgGs (or whichever Ig you are looking for) using protein A/G resin if you want to effectively 'reverse IP' it - whether from the media or from the cells directly. Elute IgG off the resin with 100mM glycine pH 2.5 (20-30uL say per 20 uL bed volume of resin), neutralize with 1M Tri pH 11 (5uL) [your sample loading buffer should be blue] then run your PAGE.
Go for it, it vl work hundred percent. I am agree with Massar that first you would optimize the Ab dilution , so that it dont give any background signal.