No, HHS does the vast majority of public health research in the US, which includes both CDC and NIH. The CDC budget is roughly a quarter of the NIH budget; most public health research is NIH-funded. The CDC is more operational- they take what is learned and apply it to prevention and treatment.
It may in fact be the case that more HHS-funded public health work is downstream of NIH than of CDC, but NIH is obviously not primarily a public health organization; here's the most recent RePORT data:
