Can I use a 3 volt DC 1 amp power source to light about 20 5mm 2 pin diodes?
For that size of diode, 10mA each should be fine, so a 1A PSU would be more than enough. Ideally use a series resistor per diode, as this controls the current in each LED individually. If you parallel all the LEDS, slight voltage drop differences between them could result in large current differences, but for a Christmas decoration that's probably not a job-stopping problem. 

A 3v supply for white LEDs is a bit low. Is it nominally 3v, so is much higher off load, or is it a regulated 3.00v?

I recently measured some chinese white LEDs that I bought in a 1000 lot from eBay, and got 1mA at 2.65v, 10mA at 3.2v. Your mileage will probably vary. If you were to connect those to a regulated 3.00v without any series resistor, you would get about 6mA per LED, which is quite reasonable, there's even some elbow-room in there for the inevitable temperature variations.

If you have a meter, measure the characteristics of the LEDs you have, and go from there. If not, whack 220ohms in series with each LED, put them on your adapter, and see what result you get. It may be OK.