Prove completeness of a language with Turing Machine
I program in Java a class Turing that takes the transitions table, the input value (to be calculated) and the initial state and simulate a Turing Machine


This is not enough. You still need to prove, that your class can simulate ANY Turing machine.

Normally to show Turing completness one will


Implement Universal Turing Machine in this language and
Show how unlimited tape is possible.


Of course unlimited tape is impossible in realistic hardware, but even for idealized translation model for languages like C, there are problems

Also I want to add one more clause: make sure that your language well-defined enough. It shall either have mathematical definition or clear formal translation model in language standard.