Final answer:
Theodore Roosevelt is most remembered for the construction of the Panama Canal, which had a major impact on U.S. foreign policy and demonstrated his 'big stick' diplomacy.
Step-by-step explanation:
President Theodore Roosevelt is most remembered for his role in the construction of the Panama Canal, an event that had significant implications for U.S. foreign policy and international trade. The strategic need for a shortcut between the Atlantic and Pacific oceans was clear, and Roosevelt was determined to see the canal built. His aggressive foreign policy, characterized by the philosophy of "speaking softly and carrying a big stick," led to the U.S.'s support for the Panamanian revolt against Colombia and the subsequent creation of an American protectorate in Panama to secure the rights to construct the canal.
This act vastly increased the U.S. impact in Latin America and showcased Roosevelt's belief in the United States' right to police the Western Hemisphere. Despite the controversy and costs involved, the Panama Canal stands as a lasting legacy of Roosevelt's presidency and his vision for American expansion and power projection.