No. This is a great question. A 2016 study by Wilding et al., from University of Michigan, demonstrates that silver does not alter gut microbiome diversity even at levels 2,000X the EPA Daily RfD.
Figure data from , Wilding, et al., (2016). “Repeated dose (28-day) administration of silver nanoparticles of varied size and coating does not significantly alter the indigenous murine gut microbiome.” Nanotoxicology 10(5): 513-520.