How do you refer to the judgment day of a single person? What do you call the day where a man is judged by God (christian God) and God deliberates whether he is to be sent to hell or not. Is it called Judgment Day or something else?
Judgement Day generally refers to the Last Judgment, or Day of Reckoning, or Doomsday, or however you want to translate Yom haDin (יום הדין).
When individuals die, they are judged on the spot, in what Catholics refer to as particular judgement.
Roman Catholic Church
The judgement of each individual soul immediately upon its separation from the body at death; opposed to general judgement.
Christians who are not Catholic don't have any such belief - thanks to Brad for the reminder - so they may have no word or phrase to describe it, unless they're referring to the Catholic concept.
And as long as I'm adding editorial notes, I'll mention that I suspect the term particular judgement may not be especially well known by non-Catholic English speakers. I, a non-Christian native English speaker, was familiar with the concept from pop culture, but had to look up the proper term, which I had probably never heard before.
I would guess that most Americans also don't know the proper term. As I suggested in a comment, in a casual context, we'd probably describe the event.
2nd Edition; There are two answers
Popular concept, and the Religious concept.
Popular concept Ref DiffSense
Apocalypse; There does not appear to be any particular uniform Christian belief of judgement separately to that arising at the time of the Apocalypse; an imminent cosmic cataclysm in which God destroys the ruling powers of evil and raises the righteous to life in a messianic kingdom.
Day of reckoning as a noun (religion): The final and eternal judgment by God of all nations; Final Judgment.
Judgement day as a noun (Christianity, Islam, Judaism): The Last Judgment, Final Judgment, final trial of all humankind, both the living and dead by God expected to take place at the end of the world, when each is rewarded or punished according to his or her merits.
Note; Spelling According to C.E.D. Judgment Day
Q. How do you refer to the judgment day of a single person?
A. There is no simple judgment day of a single person
Q. What do you call the day where a man is judged by God (Christian God) and God deliberates whether he is to be sent to hell or not. Is it called Judgment Day or something else?
A. Every single person who has died all throughout Earth’s history will be woken up by a call of Jesus but only the wicked will face a day of Judgement not everyone.
There is a misunderstanding in your idea, relating to Christian beliefs, of what happens when a person dies. Which is not really surprising as what happens is defined in the Bible but each sect of Christianity must put their own spin on this to attract members to "My Church Inc". There is really only one battlefield where the different denominations can fight, "The Arena OF Eternal Life" and what it is going to cost each of us to gain entrance.
Most of the Christian, or non Orthodox Christian, beliefs relate to what St Paul said "Jesus would decide". If you want to relate to what God says then you should relate to the Original book and I would suggest the Jews have more expertise in relation to that.
Also many commonly referred to events are not actually biblical but derived by the Catholic Church. For example "Particular Judgement" did not arrive till the 16th century, so I am not quite sure what happens to all those who died during the previous millennia. Also Everyone who has died will wake when Jesus calls but according to St Paul only those who believe in Jesus can enter into the Kingdom of heaven. So where Adam, Abraham and the likes, who have never heard the teachings of Jesus, will end up I am not sure. An exert from The Apocalypse of Peter, a text written during the early to mid second century, again relates to the judgment of non believers, but in this case of a more Jewish rooted belief "The lawlessness"
- And then God will come unto my faithful ones who hunger and thirst and are afflicted and purify their souls in this life; and he will judge the sons of lawlessness.
The Bible does not teach that we have an immortal soul that immediately goes to heaven when we die.
What Happens When You Die? You Sleep until you are woken up by the call of Jesus.
As the other answers explained, Judgement Day day refers to the Last Judgement (for Catholics and Protestants) which will happen at an unspecified time in the future for everyone at the same time.
Juhasz answer is the first time I've heard "particular judgement." I've never heard this word, and I doubt my catholic friends have either. Most protestant lay people assume you'll be judged when you die.
Since it's not nice to talk about people dying, the most common phrases I've heard is "The day he meets God/Jesus" or "The day he meets his maker." At least in America, the belief they you are judged the day you die is widespread. Even if the listener doesn't agree with the theology, they'll know what you mean. It's also not a commonly contentious point of theology, so you're very unlikely to offend people.
While technically Judgement Day is for people and the Day of Reckoning is for nations, in American English we commonly speak of a person's individual Day of Reckoning, and it implies a more personal act such as the OP is looking for.
Although such a use may be theologically incorrect, "It was his Day of Reckoning" will be easily recognized as a time of personal judgement either literally at a person's death, or metaphorically any time they are called to account for their behavior.