Where there's a will there's a way
If this art is vector it can be made to be compatible with screen printing. If it has sufficient resolution, it could even be done in Photoshop by someone skilled with alpha channels.
I did something similar for someone a couple of months ago, only it was four spot colors. Basically, I recreated the art in Illustrator and then converted it into pre-separated high resolution screened art in Photoshop.
Printing "onto fabric" is a bit vague. Do you want to produce one gross (144) black T-shirts (silkscreen) or a single polyester fabric banner (inkjet) with a black background? Will it be hanged outdoors? How long does it need to last? Did you select fabric only because it seems more durable than paper? There are many types of material like poly-cotton canvas, polypropylene or reinforced scrim.
Printing method may be based upon quantity and intended use. Silkscreening is costly to set up for one print. Inkjet printing on polyfab is possible, in which case the black background will be printed along with the logo, leaving the fabric unprinted for the white logo text. For short-term outdoor use or longer term indoor use, UV aqueous inkjet printing is fine. Longer term outdoor use may require solvent inkjet printing.