I think the added 'do' is an english thing. In german you simply say 'One can that not' (literally translated), and in danish it's the same. In both cases, the 'to do' is implied but does not have to be said out loud for the sentence to be correct. I guess we're just too lazy to say that :)
The second one should be accepted (but isn't, or at least "that one cannot do" isn't); the first is pretty unusual English, really only okay in certain circumstances.
"Can you get there by car?" "That you cannot." This is okay, but maybe overly dramatic. A more usual answer would be "No, you can't." "That you cannot do" is also okay here, but again pretty dramatic.
"I am going home." "That you cannot." This is wrong. Here you would need to say "That you cannot do," or more typically "You can't do that."