I think the objection that the "Raster to Polygon" tool is making is simply that there are too many floating point values to make polygons out of. I doubt that you actually want a separate polygon for each cell.
What I would do would be to generalize the raster into categories that make sense (you could use the cutoff values from the classification). I'd make these into integer values and then run the raster to polygon tool.
If you have Spatial Analyst, you should be able to generalize the raster using Reclassify or Con (though I think it would be easiest with Reclassify).
EDIT: To further assert that this is the problem, see the Help documentation for the Raster to Polygon tool which states (referring to the in_raster): "The raster must be integer type."