Is it possible to safely operate a Boeing 777 on diesel instead of Jet-A? Are there any regulations that prevent this? Would it cause any damage?
No. The fuel system would risk to clog up with gelled diesel fuel in the low temperatures at cruise altitude. If you had omitted "safely" in your question, the answer might be different.
Before I repeat what others have written before, I refer you to this excellent answer by @voretaq7. The essential part is:
Why don't we burn highway diesel or heating kerosene in a 747? In a word, purity. The additional controls imposed on Jet-A fuel by the associated standard produce a product with well-known properties. When you fuel a plane with Jet-A you can be confident that fuel won't freeze in the tanks at altitude (at least not unless you cold-soak it below -40 degrees). You also know the fuel will be "clean" and won't gunk up fuel filters or leave deposits inside the combustion core of the engine that can cause problems later.