Could the owner be
negligent for this second water damage to my unit?
It's POSSIBLE, but it does not sound as though you have the facts to show he was negligent.
Was it his responsibility
to have his plumbing inspected after the first incident?
It was his responsibility to do what was reasonable under the circumstances.
You didn't say whether the first repair involved opening up the ceiling and replacing part of it. If it did, then your repair contractor should have looked up there to see if he could spot the cause, or at least learn what could be learned from looking up from there.
But I'm guessing the repair only consisted of repainting, which would suggest the problem was not very serious, and it would not have been reasonable to expect the other owner to drop everything he was doing and run over there, disconnect and move the washing machine, and open up the wall that same day. Then of course the problem self-healed, so if your neighbor had come and done all that the next day or the day after that, everything would have looked fine at that time and all his effort would have been wasted.
If a condo unit owner
who is not a professional performs his own inspections, maintenance, and
repairs is he not liable and thus negligent?
Not just because of that! You would have to show that work he or his workers did was done completely improperly (not to generally accepted standards of workmanship) and that was the cause of your damage.
I called his agent and placed a formal claim against his policy for
negligence and still have not been contact by them.
You can't place a claim against anyone else's insurance. You can inform his insurance company that you believe their insured's negligence caused your ceiling damage. But the only way to force them to pay you any attention is to file a lawsuit against your upstairs neighbor.
Do you have insurance on your own unit? Are the ceiling repairs costly enough to be worth filing a claim against your own policy? If so, I recommend reading your policy to see (1) whether this would be covered and (2) how promptly they require you to contact them.