Can you sue the airline if you get sick after flying?

You can absolutely sue the airline if you get sick after flying with them.

You will lose, and if it makes it as far as the courts then you'll almost certainly need to pay their legal costs as well as your own. However that does not stop you from suing them.

This from Patrick Smith's excellent book Cockpit Confidential:

Studies have shown that a crowded airplane is no more germ-laden than other enclosed spaces - and usually less. Those underfloor filters are described by manufacturers as being of hospital quality. I needn’t be reminded that hospitals are notorious viral incubators, but Boeing says that between 94 and 99.9 percent of airborne microbes are captured, and there’s a total changeover of air every two or three minutes - far more frequently than occurs in offices, movie theaters, or classrooms.

You definitely have a right to sue anyone you like, but you might want to make sure the facts support your position before you do, to avoid embarrassing and expensive outcomes.

@Doc has already given a great straight answer to the question asked.

To expand on that: How will you know that you got sick at the plane and not at your

1. holiday destination

2. departure airport

3. arrival airport

4. any public transport taken anywhere during that day

5. anybody else you've had contact with

You will know it takes symptoms a bit to show up (for flu - one to four days). I remember my flu symptoms started appearing only when I was already home! This isn't something that you can prove to anybody that you got it from a specific location or person.