The answer is simply
of course, absolutely not
and it's surprising there is so much humming-and-hawing about it here.
It's an extremely US-centric idea, it's "just silly" if you will. Indeed the dollar symbol is often used in say political cartoons, financial cartoons, very much as a symbol of the US broadly.
If (for some reason) you wanted to use one symbol, you'd probably use the planet's main international currency, the euro, which is used by about twice? as many people as the USD, and of course in a dozen+ languages. (Of course, the Chinese Yuan is used by far more people than the Euro, and if I'm not mistaken there are now more net-users in China than anywhere else.)
(Oh - it turns out the the number of net users in China is double the US population ... again, another point suggesting it's "a bit silly" to use the dollar symbol.)
Moving on to a more positive note, what symbol should you use:
It seems quite common to use a melange of various major currency symbols:

Or, use "notes and coins" style icons ...

Or, use "bullion bars" style icons ...

There's three great choices, I think those are the main three choices open to you in developing an icon.