Try this:
> transform(df, day.prev= c(NA,diff(as.Date(df[,1]))))
V1 day.prev
1 1993-05-05 NA
2 1993-05-14 9
3 1993-06-27 44
4 1993-06-27 0
5 1993-07-10 13
6 1993-07-27 17
7 1993-08-23 27
8 1993-09-04 12
Or
df$days.prev <- c(NA,diff(as.Date(df[,1])))