R time_trans works with objects of class POSIXct

This should work fine using the argument date_labels:

library(ggplot2)
library(lubridate)
data <- read.table(text= "
Date      Closed
                   2010-07-19    0.0808
                   2010-07-20    0.7547
                   2010-07-21    0.8547", stringsAsFactors=FALSE, header = TRUE)

data$Date <- ymd(data$Date)
ggplot(data, aes(x=Date, y=Closed)) + 
  geom_line() + scale_x_date(date_labels = "%b-%d-%Y")

enter image description here

Regarding you codes, you can't use scale_x_datetime if the format of your data is date (Assuming that they are). That's why you get the error: Error: Invalid input: time_trans works with objects of class POSIXct only

If your Date column is a character vector you can convert it using as.Date inside aes or previously df$date <- as.Date(df$date) without using additional packages:

last_month <- Sys.Date() - 0:29
    df <- data.frame(
      date = last_month,
      price = runif(30)
    )
    df$date <- as.character(df$date)
# Plot
ggplot(df, aes(as.Date(date), price)) +
  geom_line()+
  scale_x_date(date_labels = "%b-%d-%Y")

enter image description here As per your desired format, inserting a line break between the month and day and year (\n inside date_labels) and rotating y-axis tick marks angle = 90 in theme:

ggplot(df, aes(as.Date(date), price)) +
  geom_line()+
  scale_x_date(date_labels = "%b-%d-\n%Y")+
  theme(axis.text.y = element_text(angle = 90))

enter image description here