Consider the given problem in which every positive integer greater than 1000 is colored either red or blue. The condition is that the product of any two distinct red numbers must result in a blue number. We need to determine if it is possible that no two blue numbers have a difference of 1.

1. Understanding the Condition for Red Numbers:\
If two distinct red numbers, say $r_1$ and $r_2$, exist, then their product $r_1 \times r_2$ is a blue number. This implies that taking any two red numbers results in producing a blue number when multiplied.

2. Exploration with a Potential Contradiction:\
Suppose for contradiction that it is possible no two blue numbers have a difference of 1. This would mean that any integer immediately following or preceding a blue number cannot be blue.

3. Deducing the Red Number Structure:\
Consider the smallest red number above 1000, denoted as $r$. Any other red number, say $r + 1$, would force $r(r + 1)$ to be blue, but as $r(r + 1) > 1000$, and $r$ and $r+1$ differ by 1, they cannot both satisfy the condition of having a blue product without contradicting the impossibility of two blue numbers differing by 1.

4. Inductive Reasoning for Blues:\
If there is a sequence of blue numbers $b_1, b_2, \ldots$ such that $b_1, b_2, \ldots = b_i$, and no two differ by 1, they effectively segment numbers. But by the infinite pigeonhole principle, within a sufficiently large range above 1000, numbers will cluster such that difference 1 can occur, primarily interrupted if numbers are red. However, had these numbers been interrupted by reds, the product becomes blue.

5. Conclusion:\
Ultimately, if only a finite or systematic pattern prevents two blue numbers from differing by 1, they lead to an inherent contradiction with the replacement into blue numbers upon multiplication constraints of red. Hence, controlling integer spread leads to unavoidable adjacent blues or complement violations through multiplication triggers for blues.

Thus, given any arbitrary assignment seeking to fulfill these conditions, attempting no two blues differing by 1 will create a contradiction. Therefore, it is not possible for this coloring system to exist such that no two blue numbers have a difference of 1.

Therefore:
$$\boxed{\text{No}}$$