To remove the x-axis labels, text, and ticks, add the following function to your plot:
theme(axis.title.x=element_blank(), axis.text.x=element_blank(), axis.ticks.x=element_blank())
Here element_blank() is used inside theme() function to hide the axis labels, text, and ticks.
In your case:
ggplot(data = diamonds, mapping = aes(x = clarity)) + geom_bar(aes(fill = cut))+ theme_bw()+ theme(axis.title.x=element_blank(), axis.text.x=element_blank(), axis.ticks.x=element_blank())
Output: