Not exactly. However, assuming that we will eventually be able to reverse the effects of aging, cancer formation will become much more rare as a by-product. The explanation for this is a bit involved, due to the complexity of the subject.
One of the most fundamental processes that drive human aging is the irreversible accumulation of genomic (and mitochondrial) DNA mutations. The reason for this is simply because it is extremely unlikely for DNA damage to get repaired by chance; DNA is the most complex molecule in any organism, and once information in it is lost for good, there is little hope that it will be reconstructed by random chemical reactions alone.
Other important forms of aging-associated degeneration include atherosclerosis, the formation of advanced glycation end-products, the accumulation of indigestible waste products within lysosomes as well as proteinopathies (although these can be caused by DNA mutations; see below) such as Alzheimer's disease. Telomere shortening can also be considered to be a form of aging-associated degeneration, but it is almost certainly not a central mechanism underlying any of the more noticeable aspects of the symptomatology of physiological aging due to its easily reversible nature (i.e. otherwise we would already have seen "immortals" appear due to random germline mutations causing overexpression of telomerase within stem cells).
While the accumulation of genomic DNA mutations eventually leads to carcinogenesis, cancer is actually a relatively unlikely consequence of this process, since multiple key mutations in several oncogenes/tumor suppressor genes tend to be necessary for transformation. As such, a much more likely (initial) consequence of the accumulation of DNA mutations would be the random inhibition of cellular functions by knock-out and modification of actively transcribed genes (or random activation of silent genes). This would explain why in old age, the body tends to be 'worn out' in general; if nearly all cells have some degree of dysfunction, we would naturally expect to see the body as a whole to perform poorly as well.
Also, even when enough mutations have accumulated to potentially cause cancer, usually, this ends up triggering the anti-cancer defenses of a cell and turn them senescent. When stem cells become senescent, they stop dividing. This would neatly explain why wounds and infections heal much slower in older individuals. In fact, it has been shown that hematopoietic stem cells lose their regeneration capacity over time, and that this is correlated with the accumulation of DNA mutations (Rossi et al. 2007).
In addition, random DNA mutations can also conceivably trigger proteopathies (e.g. Alzheimer's disease or idiopathic prion disease), since for many of these you only need one gene producing a 'bad version' of a protein to cause aggregation and cell death. Both prion proteins and amyloid-beta tend to be infectious (Jaunmuktane et al. 2015), so it is entirely possible that a single mutation in a single cell could trigger large-scale degeneration.
So, to stop aging, you would at least need to fix DNA damage periodically, and this would more or less prevent cancer as a by-product. However, I imagine it would be hard to repair the DNA damage in each and every cell in the body with a single treatment (e.g. with some kind of a viral correction therapy where the treatment is administered by intravenous injection, it would be extremely unlikely for the therapeutic agent to reach all cells every time, since there is a limit to how concentrated you can make it). As such, it would still be possible for a cancer to arise at some point, and in this sense, it is entirely possible for us to end up in a situation where we are able to reverse all aspects of biological aging, and yet people still die of aggressive cancers every now and then.
References:
Jaunmuktane, Zane, et al. "Evidence for human transmission of amyloid-β pathology and cerebral amyloid angiopathy." Nature 525.7568 (2015): 247.
Rossi, D.J., Bryder, D., Seita, J., Nussenzweig, A., Hoeijmakers, J. and Weissman, I.L., 2007. Deficiencies in DNA damage repair limit the function of haematopoietic stem cells with age. Nature, 447(7145), p.725.
