You can do this with natural light if you have a nice big window. For a sleeping newborn, you don't need a very fast shutter speed, but if he/she is thrashing around at all, you should keep your shutter speed at least to 1/125. I'd keep the aperture as wide as possible (f/1.8) and focus on the eyes. Shoot in RAW, if you can, then in post editing, you can adjust the Black tool to darken the background. Keep the room as warm as possible, bring in a space heater and warm up the blankets you plan to use. When I shoot newborns, I wear shorts and a t-shirt even in the middle of winter. My living room is that warm.
You can do a white background too. This shot used a white sheet. I used the dodge tool in photoshop elements to blend away any wrinkles or grey spots. Again, this was natural light on the back porch. To get a true "high key" white background, you would need a couple lights lighting the sheet to competely blow it out. This shot still a bit of a blue hue to the background.
If you have a grey card, use it to set the white balance. Newborn skin tends to be a little pinkish or yellowish and the camera rarely makes the right choice on its own.
I think black is easiest to work with and more forgiving of lighting errors. If you have a flash, you could bounce it off a white ceiling to help with lighting. Never shoot the flash directly at the baby...it will be too harsh. Good luck. Share them with us, please!

...and congratulations.