Final answer:
FDR did not support the federal antilynching law due to the need to maintain the support of Southern Democrats for his New Deal programs, as well as historical precedents that favored states' rights and local jurisdiction over such matters.
Step-by-step explanation:
FDR chose not to support a federal antilynching law primarily because of the political implications it would have on his presidency and the legislative agenda he pursued. Despite private support for the legislation, he was wary of alienating Southern Democrats, whose support was crucial for his New Deal programs. Intervention in local issues like lynching was avoided by previous presidents, highlighting the delicate balance between federal authority and state jurisdiction. This hesitancy can be traced back to historical concerns over states' rights and the legislative obstacles faced, with a context deeply rooted in racial discrimination and political pragmatism.
Furthermore, the continued opposition by white Southern Democrats, who feared that such legislation would upset the racial status quo and bolster federal power over state sovereignty, presented a seemingly insurmountable barrier. Roosevelt's administration, aware of the backlash that might ensue, tread carefully to preserve the political coalition vital for the success of the broader New Deal efforts.