diff operates on files, not on content passed in parameter. Command substitution ($(...)) replaces the command with its content, this is not the tool you need.
What you actually need is what is called process substitution (<(...)), which replaces the command with a pseudo-file (actually a pipe) that contains (actually conveys) the result of the command, which is executed in a sub-process.
diff <(curl "https://raw.githubusercontent.com/dockerfile/ubuntu/master/README.md") \
     <(curl "https://raw.githubusercontent.com/dockerfile/ubuntu/master/Dockerfile")
Unfortunately, as you added in note, this is a bash extension that is not known to POSIX shells (/bin/sh). To my knowledge, there is no solution with /bin/sh and you have to save at least one of the two curl outputs in a file:
$ tmp=$(mktemp)
$ curl -o "$tmp" "https://raw.githubusercontent.com/dockerfile/ubuntu/master/README.md"
$ curl "https://raw.githubusercontent.com/dockerfile/ubuntu/master/Dockerfile" \
  | diff "$tmp" -
$ rm -f "$tmp"