I've gone down with the flu. I think it's not life-threatening ,so I don't need to go to a clinic.

Is that natural?

Question

I've gone down with the flu. I think it's not life-threatening ,so I don't need to go to a clinic.

Is that natural?

Answer

I've come down with the flu. I don't think it's life...

You would generally say you've come down with the flu or a cold. 
Or that you've caught the flu or a cold.

I would say 
I've come down with the flu/I've caught the flu, but it's not bad so I don't need to go to the doctor.