I'm no big fan of Gresham's law but it has some merit.
What you say next is correct, but its not because of the Gresham law. Gresham law only applies if there is an artificially fixed exchange rate between two currencies. If the currencies float freely Gresham law does not apply. F.e.: The market price of gold is now over $1540, so it means there are people buying and selling around that price. If the government now sets a price of $800 by law, nobody will sell it at that price and gold will go into hidding. This is Gresham law. But if the price is freely floating, if you raise the price enough you will find someone willing to sell you gold. When prices are free Gresham law does not apply. It only applies when there is a artificially fixed price.
(Btw, this confussion about the Gresham law is not strange, because the keynesians have tried to misrepresent the Gresham law so they could criticize their mischaracterized version)
The reason what you say is true, is because the government discourages the use of any alternative currency by forcing everyone to pay taxes with the government currency and by legal tender laws, thus creating a monopolly.
But I am with you that Bitcoin has the potential to overcome all this.
As long as you can pay with fiat currency for everything, you don't have a high incentive to pay with BTC. On the other hand, I really like spending BTC because its so easy and secure compared to other forms of online payment.
I regularly buy a few BTC and spend them on things I would have bought anyway or make some donations.
I know that many merchants who accept BTC today instantly convert them back to fiat again, so one could say it is not really worth the hassle but you have to keep in mind that we're bootstrapping a decentralized global currency here - that's not an easy task! It is a bit like voting for a politician who you think won't win the election - some could say your vote was lost but it is very important for a democracy that everybody votes for the one he believes to be the best.
The good thing with Bitcoin is, that it has enough advantages on its own to flourish regardless of everyone acting for its greater good. Bitcoin's value (ie the demand for them) can come from at least two sources: Bitcoin as a medium of exchange and Bitcoin as a store of value. If they are perceived more as a good store of value then there will be more hoarders and their demand will raise the price. At the same time, less BTC in circulation will limit their usefulness as a medium of exchange which lowers the demand from merchants hence lowering the price again. This in turn will make some hoarders part with some of their coins because they fear that they are not so good a store of value after all -> more BTC in circulation, more attractive to exchange them... Rinse, repeat.
I really see such positive feedback loops built into the foundation of Bitcoin as its greatest quality. Despite all the flaws, imbalances and imperfections of human society and people's motives - they just work!
You probably can tell that I'm in an optimistic mood today