Because the medical profession was opposed to it. Roosevelt's administration feared that including the universal health insurance provisions would kill the entire Social Security Act.

For the sake of passing the Social Security bill, we postponed the introduction of the bill on health insurance as the opposition was so great from the American Medical Association (principally) that it would have killed the whole Social Security Act if it had been pressed at that time. Lampman, Robert J. "Social Security Perspectives: Essays by Edwin E. Witte." Madison, University of Wisconsin (1962).

Edwin Witte was the Executive Director of the Committee on Economic Security, established in June 1934 to work on what would become the Social Security Act. The proposals for providing public health insurance provoked outrage from medical organisations such as the American Medical Association

By then, the AMA's opposition to national health insurance had become even more fierce. in the Journal of the American Medical Association, national health insurance was presented as an un-American menace to the public health, undermining the traditional American values of individualism and self-reliance ... In later years, Edwin Witte would claim again and again that it had been his "original belief" that health insurance was not politically feasible, mainly due to the opposition of the medical profession. - Kooijman, Jaap. ... And the Pursuit of National Health: The Incremental Strategy Toward National Health Insurance in the United States of America. Vol. 8. Rodopi, 1999.

In early 1935, the Social Security bill was languishing in Congress, where the universal healthcare section received little vocal support. In contrast, the AMA was strongly opposed and influential in Congress. This convinced Witte and the CES that retaining public health insurance would prevent the bill's passage.

CES staffers admitted privately that ... "extreme care is necessary to avoid the organized opposition of the medical profession," and that "there is not a very great chance for the adoption of legislation at this Session on the subject." When the time came to present the committee's final report ... those who feared that controversy over health insurance would doom the whole bill won out and the health title was dropped. - Gordon, Colin. Dead on Arrival: The Politics of Health Care in Twentieth-Century America. Princeton University Press, 2009.

With Social Security seemingly bogged down, Roosevelt decided to hold off on public health insurance, and ultimately shelved the idea altogether in favour of increasing the availability of medical care.