Mathematician here. If you mean to say that you cannot constructively prove "the real numbers are not countable", then you're wrong. As a rule of thumb, you can usually prove negative statements constructively as you would prove them classically.
A constructivist would probably state the result more positive (and stronger, constructively): To every countable set M of real numbers, there is a real number not contained in M.
A constructivist would probably state the result more positive (and stronger, constructively): To every countable set M of real numbers, there is a real number not contained in M.