"This coin is fair" is weak in an important way as a falsifiable proposition, but for the most part is as strong as any scientific conclusion as far as falsification goes.
The one sense in which it is weak is "fair" might generally be interpreted as a value judgement. However, let me assume that here "fair" means that the coin was constructed as are typical coins from the U.S. mint and has not been modified from that constructed form by other than the same wear and tear issues experienced by the vast majority of coins from the US treasury. This is a technical definition of fair, theoretically falsifiable by someone who monitors the coin 24/7 from its creation until now.
Actually, the above paragraph shows how "fair" can be made falsifiable. Interestingly, it does not require an infinite number of tosses of the coin or even a single toss of the coin. It involves knowing a nearly complete history of the coin from its creation until now.
Now can a coin be determined to be "fair" or not by tossing it repeatedly and analyzing the results of that? If it is unfair in a straightforward way (its tosses have time-stationary statistics and a mean rate of heads different from 50%), then this will be detected with arbitrarily high confidence in a finite length of time. But if it is "unfair" in a more subtle way (for example, it is a voice-activated machine which is inert and therefore "fair-like" most of the time, but can be activated to be unfair by some mechanism), then it can not be determined to be unfair if its unfairness is not activated during its testing.
My conclusions:
- "fairness" as I define it here (which I think is pretty close to how we intuitively use it) is in principle falsifiable.
- Only by making a probably-unfalsifiable claim of time-stationarity can a "fair" coin be determined to be fair with high confidence, by tossing it. A triggerable unfairness can never be ruled out by tossing it.
- The test for "fairness" is more properly complete knowledge of the coin's history than anything as limited as tossing it a bunch of times.