No. Not even close. At best, a longbow could launch an arrow with 60 meters per second velocity. If we can assume that there isn't a very, very high tower for him to stand on and so both he and the target are at sea level, an archer shooting at the best inclination of 45 degrees will give the arrow a range of 367 meters. This is also before we start factoring in air resistance, which for a long wooden thing with feathers would be significant. As the Channel at its narrowest is 32 kilometers, the archer would have to swim out and keep swimming, re-shoot the arrow 87 times.

EDIT The ones tested in the above linked website were around 50lbs. Consulted Oakeshot, and did a bit more digging online: although it seems longbows often drew more than 100 lbs, and and their velocity might reach 100 meters/second , their range was still safely below 400 meters. In a vacuum, the range would be a bit more than 1,000 meters, but in air, to use the aerodynamic adage, "that which draggeth, falleth".

In short, no. The English channel is 35km at the narrowest point. The current world record for the longest shot with any sort of bow is 1871.84 meters (Harry Drake, 1988) with a modern crossbow. I will leave it to someone else to give information on historical bow ranges, but they will be much less than that.