How to format date output with spaces as variable in script?

The reason your example fails is because of the way the shell's word splitting works. When you run "$($nice_date)", the shell is executing the date command with two arguments, "+%Y-%m-%d" and "%H:%M:%S". This fails because the format string for date must be a single argument.

The best way to do this is to use a function instead of storing the command in a variable:

format_date() {
  # echo is not needed
  date "+%Y-%m-%d %H:%M:%S" "$1"
}
format_date
format_date "2015-09-17 16:51:58"
echo "$(format_date) [WARNING] etc etc"

If you really wanted to store the command in a variable, you can use an array:

nice_date=(date "+%Y-%m-%d %H:%M:%S")
# again echo not needed
"${nice_date[@]}" "2015-09-17 16:51:58"

For more details on the complex cases of storing a command in a variable, see BashFAQ 050.

I agree a function is the best way to go. As an alternative simply store the format as a variable rather than the whole command:

$ nice_date='+%Y-%m-%d %H:%M:%S'
$ echo "$(date "$nice_date") [WARNING] etc etc"