It is not entirely true to say that China 'declared war on the United States' in 1900, nor is it true to say that the United States did not respond.
With regards to the declaration of war, it must be noted that the declaration of war against foriegn powers was a unilateral action by the Empress Dowager Cixi, and was not broadly acknowledged. The decline of the regular, structured bureaucracy in the wake of the Taiping Civil War had left political power divided between a central government increasingly controlled by Manchu princes on the one hand, and provincial governments and armies dominated by regional strongmen on the other. In particular, what Pamela Crossley terms the 'Beiyang Intendancy' of Li Hongzhang and Yuan Shikai, based in northern China, actively vied for authority with the Manchu court. The army, or perhaps more correctly armies, were broadly anti-Boxer in terms of their leadership, and despite Cixi's declaration of war they did little to support the pro-Boxer court. While some units in and around Beijing were essentially roped into the fighting, they did so reluctantly, generally failing to apply pressure against the foreign expeditions and at one point even declaring a prolonged ceasefire during the siege of the Beijing Legations. Yuan Shikai's Beiyang Army had been recently relocated to Shandong, and he was heavily involved in keeping the Shandong Boxers suppressed by nonmilitary means even while the Qing court was backing the Zhili Boxers against the foreigners. One story has it that Yuan was present at a Boxer ritual, and after witnessing a Boxer master allegedly stop a bullet by using a blank gun fired by a stooge, drew his own pistol out and shot the master himself in order to discredit the Boxers and disperse their adherents. Similarly, the Western powers recognised that the Boxer problem was, for the most part, localised to North China, particularly in Shandong and Zhili provinces, and so the other provinces of the Qing empire were generally not targeted – the exception being the Manchurian frontier, which saw significant Russian incursions.
On the flip side, the United States most certainly did not stand by while its legation quarters were besieged and its soldiers threatened by the Boxers and the court. U.S. Marines formed a small but notable contingent in the early months of the fighting around Tianjin, and the American government did what it could to scrounge up troops to support the allied campaign. Future Army Chief of Staff Adna Chaffee was pulled out of Cuba and given command over American forces in the conflict, with orders to coordinate but not necessarily cooperate with the forces of the other major foreign powers. Meanwhile, the 14th and 15th Infantry Regiments were supposed to be pulled out of the anti-guerillla campaign in the recently-conquered Philippines, but a rather bleak assessment of the situation by Major General Arthur MacArthur (father of Douglas) led to only one regiment, the 9th Infantry Regiment in Luzon, being relocated for the China campaign. In the event, the United States was a key signatory of the 1901 Boxer Protocol, which stipulated that it should recieve a 7.32% share of the 450 million tael (18,000 metric tonnes of silver) indemnity – equal to the Italian share – and the U.S. was also the first Protocol signatory to remit part of the debt in 1911, albeit specifically to fund Chinese students planning to study overseas in America, rather than directly giving the money back to the Qing.
Sources, Notes and References
Joseph W. Esherick, The Origins of the Boxer Uprising (1986)
Paul A. Cohen, History in Three Keys: The Boxers as Event, Experience and Myth (1997)
David J. Silbey, The Boxer Rebellion and the Great Game in China (2012)
Patrick Fuliang Shan, Yuan Shikai: A Reappraisal (2018)