Summary: Yes it can, but not the way you might think.
Humans aren't under a lot of selective pressure on a large scale right now. We have a lot of control over our environment, and you're accurate that disease (or I'd add massive, fatal natural disasters) are the biggest ways we would see a large genetic shift. (Or genocide, which is an extremely unnatural, artificial selection, but some of our species have historically been horrific enough to try to do this.)
Small local natural selection has happened in areas where Malaria is endemic: many people die from malaria. some people die early from homozygous sickle cell anemia, but those who are heterozygous have "sickle cell trait" (because of incomplete dominance) and have deformed enough RBC that malaria isn't able to infect them well. So the survival advantage is enough that the prevalence of sickle cell alleles in these areas is higher. There's a similar story with Glucose-6-phosphate dehydrogenase deficiency.
But on a scale you might be thinking of, the primary mechanism would be a global pandemic infection that targets people with a specific trait. For example, if suddenly an incurable contagious, virulent, highly pathogenic, and fatal virus/bacteria that was only fatal to those with an allele for blood type A were to emerge. Theoretically it would cause a genetic shift to eliminate type A blood from the gene pool, and reduce prevalence of type B (by eliminating AB due to the A). Or if it attacked only one location of the world, but that's highly unlikely due to global travel.
Medicine could theoretically catch it in time - mitigate impact and prevent a large shift by creating a vaccine or cure that is effective and easily mass-produced. The most likely to succeed would be a vaccine (yay for public health!).
Variations on this have happened to reduce or eliminate pathogens (such as polio and smallpox because of the vaccine) and their effect on mankind, but those have little to do with selection for human traits that would cause genetic shift.