No, the protocol would still work. But competition would be easier. If the underlying value of bitcoin--trustless, decentralized, limited supply, etc--is duplicated, then bitcoin would have competition. That's not necessarily a bad thing, but it's not a likely thing.
Discussion
...it's not a likely thing because for the same free cost of electricity you could mine bitcoin (established market and existing high "price point") or some new market entrant. Cost v benefit, bitcoin wins.