Which of the following statements is definitely FALSE? 

From the clue given that a player who faces the table does not sit adjacent to someone who also faces the table, we understand that the directions alternate for the players around the round table. The configuration would look like, 

Now we are given that only European players face the table, and that seed 1 and 2 are facing each other. That means seed 1 and 2 are European. It is also given that pairs sitting opposite to each other, have their sum of seeds add up to a prime number. 

Before doing trial and error, we look at the next clue that says that seeds 3 and 8 sit on either side of seed 1, so that means seed 3 and 8 are not Europeans and 3 and 8 are not sitting opposite to each other. 

Now that 3 and 8 cannot be a prime number sum pair, we look at the only other possible option of pairs. 
1 and 2 (given)
3 and 4
5 and 8
6 and 7

We have the following arrangement using the clues thus far, 

Now we look at the clue that says that the player to the right of the second seed is not a top 4 seed. So, we see the possible configurations, we observe that either 4 or 5 can be placed adjacent to seed 2, that means seed 5 is to the right of seed 2. And the second part of that clue says, the player to the left of that player is not bottom 2. To the left of the 5th seed (considering the fact the player is facing away from the table) is either 6 or 7, since it is said not bottom 2, it is 6th seed. And using the prime number configuration we can find the remaining two players as well. 

The final configuration using all the clues is as follows. 

We see that 6 does not sit adjacent to 4.