Final answer:
The false statement about the president's role as chief diplomat is that the president can declare war without congressional approval. The Constitution actually grants Congress the power to declare war, not the president.
Step-by-step explanation:
The role of the president as chief diplomat involves several key responsibilities in the realm of foreign policy. Among these duties are the authority to negotiate and sign treaties with other countries (with Senate approval), appoint ambassadors to foreign nations (also with Senate confirmation), and recognize foreign governments. However, one statement about the president's role as chief diplomat is false.
While the president has several powers in foreign relations, the statement that 'The president can declare war without congressional approval' is FALSE. The Constitution grants Congress the power to declare war, although the president does have significant military powers as commander-in-chief.
Therefore, the correct answer to the original question is: c) The president can declare war without congressional approval is the false statement about the president's role as chief diplomat.