Well, you have to understand it was the Jews that had rejected what God had given them and fell into a abomination that caused the desolation. Antiochus, as already made clear, doesn't fit, as Jesus declared it looking forward in time, not backwards.
Now you are correct in comparing Daniels time of the destruction of the first temple. Christ points us to Daniel, and we see in Daniel 11:31: “31 And arms shall stand on his part, and they shall pollute the sanctuary of strength, and shall take away the daily sacrifice, and they shall place the abomination that maketh desolate.” Daniel 11:31(KJV)
What this shows as that the issue is tied to the relationship between God and His people and the accompanying worship to God, their rejection first of this, being contrary to the worship God deserves which caused the ruin of that nation. We see what it was in a warning given them previously:
Leviticus 26 King James Version (KJV)
1 Ye shall make you no idols nor graven image, neither rear you up a standing image, neither shall ye set up any image of stone in your land, to bow down unto it: for I am the Lord your God.
2 Ye shall keep my sabbaths, and reverence my sanctuary: I am the Lord.
3 If ye walk in my statutes, and keep my commandments, and do them;
4 Then I will give you rain in due season, and the land shall yield her increase, and the trees of the field shall yield their fruit.
5 And your threshing shall reach unto the vintage, and the vintage shall reach unto the sowing time: and ye shall eat your bread to the full, and dwell in your land safely.
6 And I will give peace in the land, and ye shall lie down, and none shall make you afraid: and I will rid evil beasts out of the land, neither shall the sword go through your land....
...14 But if ye will not hearken unto me, and will not do all these commandments;
15 And if ye shall despise my statutes, or if your soul abhor my judgments, so that ye will not do all my commandments, but that ye break my covenant:
16 I also will do this unto you; I will even appoint over you terror, consumption, and the burning ague, that shall consume the eyes, and cause sorrow of heart: and ye shall sow your seed in vain, for your enemies shall eat it.
17 And I will set my face against you, and ye shall be slain before your enemies: they that hate you shall reign over you; and ye shall flee when none pursueth you.
18 And if ye will not yet for all this hearken unto me, then I will punish you seven times more for your sins.
19 And I will break the pride of your power; and I will make your heaven as iron, and your earth as brass:
20 And your strength shall be spent in vain: for your land shall not yield her increase, neither shall the trees of the land yield their fruits.
21 And if ye walk contrary unto me, and will not hearken unto me; I will bring seven times more plagues upon you according to your sins.
22 I will also send wild beasts among you, which shall rob you of your children, and destroy your cattle, and make you few in number; and your high ways shall be desolate.
23 And if ye will not be reformed by me by these things, but will walk contrary unto me;
24 Then will I also walk contrary unto you, and will punish you yet seven times for your sins.
25 And I will bring a sword upon you, that shall avenge the quarrel of my covenant: and when ye are gathered together within your cities, I will send the pestilence among you; and ye shall be delivered into the hand of the enemy.
26 And when I have broken the staff of your bread, ten women shall bake your bread in one oven, and they shall deliver you your bread again by weight: and ye shall eat, and not be satisfied.
27 And if ye will not for all this hearken unto me, but walk contrary unto me;
28 Then I will walk contrary unto you also in fury; and I, even I, will chastise you seven times for your sins.
29 And ye shall eat the flesh of your sons, and the flesh of your daughters shall ye eat.
30 And I will destroy your high places, and cut down your images, and cast your carcases upon the carcases of your idols, and my soul shall abhor you.
31 And I will make your cities waste, and bring your sanctuaries unto desolation, and I will not smell the savour of your sweet odours.
32 And I will bring the land into desolation: and your enemies which dwell therein shall be astonished at it."
At the time of Daniel they had what Solomon built for God what’s probably been the most magnificent sanctuary on earth. Yet Solomon also added worship places for his wives’ false gods in the start of what the Bible refers to as an “abomination.”
"For Solomon went after Ashtoreth the goddess of the Zidonians, and after Milcom the abomination of the Ammonites."1 Kings 11:5.
"Then did Solomon build an high place for Chemosh, the abomination of Moab, in the hill that is before Jerusalem, and for Molech, the abomination of the children of Ammon." 1 Kings 11:7.
Putting another god in place of the true God is abominable, it is a detestable act which leads to desolation when Gods protecting power is withdrawn. When Daniel wrote about the “abomination of desolation,” the first sanctuary or Solomon's Temple in Jerusalem had been destroyed in the attack of the Babylonians. But Daniel also wrote about the sanctuary being restored which was fulfilled by the rebuilding of the second temple, and they could again come and worship God.
But because of their sin and with the rejection of the Son of God, Christ the Messiah, they again reached a level of "Abomination" that was detestable to God. It led to the destruction of the Second Temple and thus the “Desolation” is the result of what you get without God.