I have a different answer for each one of you.
Ducky, I would use a fusible estabilizer before stitching. Cotton doesn't have the stability of Aida, linen or evenweaves and your stitches may be a bit loos without a stabilizer. Iron something on the back before.
Beachlady, if your piece is stitched on Aida or any other "normal" cross stitch material, you can use it in the center of a quilted wallhanging with no further treatment. Your doubts will come in when is time to quilt it. Do you quilt over the cross stitched area or not. I have tried both ways. I did not like it when I didn't stitch over the cross stitch. It looked like it had a bump in the middle. There are two things you can do. You can use a matching very fine thread and outline the major parts of the cross stitch, or you can use invisible nylon thread or a matching monofilament and stipple over the entire piece. either one of the two works well for me, depending on the piece.
I want to see both your pieces.
Maria