Final answer:
Catherine the Great considered the United States dangerous and refused to extend diplomatic relations due to fears of the spread of revolutionary ideas.
Step-by-step explanation:
The country that Catherine the Great considered so dangerous that she refused to extend diplomatic relations to was the United States. At the time of Catherine's reign, the concept of a republic that had overthrown its monarchy, as the United States had done, was seen as a radical and potentially destabilizing example for other nations. Therefore, countries like Russia, which was ruled by Catherine the Great, viewed the United States with suspicion and concern about the spread of revolutionary ideas.