This is something pretty much everyone has done. The act of throwing but also holding onto a sheet or blanket or beach towel to spread it out in the air so that you can lay it flat on the bed, ground, or whatever surface. I don't think there is a word for this. Whenever I've asked this question people just zone out with a blank expression on their face. words like open, unfold, unfurl, throw, blanket just don't seem enough. In my mind, the word spread comes closest, but I feel it still isn't quite right.
Spread is the most suitable word.
The Ngram Viewer doesn't find anything for unfurl or unfold a blanket, but it has many hits for spread a blanket.
By the way, the word bedspread is synonymous to blanket, which shows the suitability of the verb "spread" to talk about spreading a blanket.
A single verb can describe sequential actions whose individual descriptions are usually unnecessary in common speech, but might be employed in vivid storytelling or a work of highly mimetic fiction, especially when the sequence is interrupted.
For instance, opening a door requires an approach, reaching for the knob/handle/whatever, activating this device, stepping back while holding the knob, etc.
If something interrupts the sequence, one might say
I had just begun to open the door when …
Or in a more vivid narrative
My fingers had barely grazed the door handle when …
Spreading a sheet on a bed or a towel on the sand is a similarly sequenced action where the airborne portion usually doesn't need to be described, mainly because except in the narrative situations I mentioned, it simply isn’t topical. Thus the blank stares.
So what about vivid narrative?
The isolated action is similar to what you do after lying on the beach towel until sunburned: you shake out the towel to get rid of the sand, or if you couldn’t resist eating crumbly cookies/biscuits in bed, you’d shake out the top sheet to get rid of the results. You perform the same action on a rug held out the window or to fluff out a towel you’ve dried on a clothesline, though in these contexts, the action is usually repeated a few times. It’s a kind of whipping motion you would rarely have to describe further.