Roosevelt sent Louis Johnson to India to get the Indian National Congress on side in the war against the Axis powers. The Indians wanted a say in Defence matters but Churchill was adamantly opposed to this. Strangely enough Johnson came up with a formula- one in which an Indian was put in charge of 'Defence'- i.e. mobilising resources for hostilities- while the Brits retained tactical and strategic command of the War effort- which the Indians were prepared to accept and to which Cripps, the British negotiator, was personally inclined.
However, the next day, Churchill signalled his displeasure at Johnson's actions to Harry Hopkins, FDR's representative in London. Hopkins disowned Johnson. Cripps was told to reject his formula. This meant that the Congress party was frozen out of the war effort. The Muslim League and the Communist party gained in legitimacy. Partition became inevitable. Why? Because FDR chose Hopkins over Johnson. Churchill reinforced FDR's decision, or indecision, in this respect by painting a picture of the 'loyal' Muslims of the Punjab, who were stupid soldiers, as against the seditious Hindus who were wily attorneys but arrant cowards. Churchill falsely claimed that the Muslim Punjabi was the bedrock of the Indian Army and had to be placated. Later, Wavell painted a picture of the Hindu majority provinces as on the brink of a grass roots Revolution and suggested the evacuation of the White population through the Muslim provinces of what would become West Pakistan. Since the Muslim Punjabi would need money and guns to keep the Afghan at bay- and since Partition would cut off monetary help from the rest of the sub-continent, Pakistan would need American help- as indeed transpired. Moreover, its Islamic identity, albeit maintained by subventions from the West, would make it useful in opening a back channel to the resource rich and strategically located Muslim countries to its West.
Thus FDR, by failing to back Johnson at the crucial moment, decided the future course of American policy in the region.