There are arguments made that handshakes in healthcare settings are not safe; for example see Sklansky et al 2014.
Some of the issues increasing the risk include noncompliance with hand-washing protocols (by both healthcare workers and patients) and pathogens that are less susceptible to hand sanitizers.
A fist bump with a glove coated in E coli transferred only about 10% of the bacteria as a handshake (Mela & Whitworth, 2014). Fist bumping produced fewer bacterial colonies than handshakes in a hospital environment where someone with clean hands walked around either fist-bumping or hand-shaking and then wiped their hand on a culture plate (Ghareeb et al, 2013). However, both of these studies looked at potentially infectious bacteria present: neither addressed actual infections in actual patients.
Problems with banning the handshake in favor of fist bumps may be reduced attention to hygiene (which would blunt the effect) and impacts on how patients view their physicians: they might see a fist bump as less professional and no physical social greeting might reduce trust (Reilly et al 2016).
However, as reviewed in Reilly et al, the evidence in favor of banning handshakes is a bit...shaky. Clearly handshakes are not perfectly safe, but neither is anything. I did not find any research nor is any cited in any of these papers that directly quantifies the risk to patients from handshakes (only that it is not zero), nor any that put it in context of all the other risks of pathogen spread in the environment, nor any that shows an impact on actual infections after a handshake-reducing intervention. You would have an (approximately?) 100% reduction in risk of shark attack if you didn't swim in the ocean, but by itself that is not a great argument to stay out of the water.
Ghareeb, P. A., Bourlai, T., Dutton, W., & McClellan, W. T. (2013). Reducing pathogen transmission in a hospital setting. Handshake verses fist bump: a pilot study. Journal of Hospital Infection, 85(4), 321-323.
Mela, S., & Whitworth, D. E. (2014). The fist bump: A more hygienic alternative to the handshake. American Journal of Infection Control, 42(8), 916-917.
Reilly, J., Currie, K., & Madeo, M. (2016). Are you serious? From fist bumping to hand hygiene: Considering culture, context and complexity in infection prevention intervention research. Journal of infection prevention, 17(1), 29-33.
Sklansky, M., Nadkarni, N., & Ramirez-Avila, L. (2014). Banning the handshake from the health care setting. JAMA, 311(24), 2477-2478.