How do different denominations defend their interpretations of Daniel 2's prophecy?

@Dottard answered the first part of the question, the widespread agreement on the identification of the kingdoms with the parts of the statue. I will tackle the second part: a partial refutation of postdiction.

Antiquity. I do not believe that there is sufficient archeological evidence to prove that Daniel was written before the Medo-Persian and Greek empires, though I believe that to be so. However, the Dead Sea scrolls contained fragments of Daniel, showing it to be - at the latest - written in the late second century BC, before Rome conquered the Mideast.

See https://digitalcommons.andrews.edu/cgi/viewcontent.cgi?article=1541&context=jats

Two Legs. The details concerning Rome are more profound than most people realize. Like two legs, Rome was ultimately divided into the Eastern Roman Empire (Byzantium) and the Western Roman Empire. Later on, it splintered into many brittle pieces, like the iron mixed with clay described in the prophecy.

Ten toes. The number ten symbolizes human (as opposed to divine) completeness, so the ten toes represent all the countries that split off from that empire. That list of coutries is many more than ten, but what if you only count those with overseas colonial empires?

• Portuguse Empire
• Spanish Empire
• French Colonial Empire
• Dutch Empire (Netherlands)
• Danish Empire (Denmark, which submitted to the Holy Roman Empire briefly)
• German Empire
• Belgian Empire
• Italy (in many incarnations: Venice, Florence, etc)
• British Empire
• United States of America (with peoples drawn from all the other European empires)

The above list of ten empires excludes Russia and Sweden, which were never part of the Roman Empire, and some Eastern European empires, which had no overseas possessions.

Between 1492 and 1914, Europeans conquered 84 percent of the globe (https://www.foreignaffairs.com/articles/europe/2015-10-07/how-europe-conquered-world#:~:text=Between%201492%20and%201914%2C%20Europeans,This%20was%20not%20inevitable.) Put another way, they conquered or extended influence over every nation except Saudi Arabia (technically held by Britain during the mandatory period), Japan, Korea, Thailand and Liberia. Even with these, the US occupied half of Korea during the Korean War and afterwards and had influence in Liberia. And after the US defeated Japan in WWII, the US imposed Democracy upon them and denied them the ability to reconstruct parts of their military.

See http://cdn1.vox-cdn.com/assets/4659723/Screen_Shot_2014-06-23_at_5.07.38_PM2.png

In Revelation it says:

The ten horns you saw are ten kings who have not yet received a kingdom, but who for one hour will receive authority as kings along with the beast. (Revelation 17:12)

Daniel says this about the ten kings:

23 He gave me this explanation: ‘The fourth beast is a fourth kingdom that will appear on earth. It will be different from all the other kingdoms and will devour the whole earth, trampling it down and crushing it. 24 The ten horns are ten kings who will come from this kingdom.' (Daniel 7:23-24)

Thus you see that Daniel's prophecy declares that offshoots from the fourth kingdom would be given great authority over the world. Those conquests would not begin until two thuosand years after Daniel lived. So no postdiction here!

Just to add to the other answers:

Modern science does not accept the supernatural. That includes Theological Departments at independent universities. See - Critical Scholars. So, they deny the miracles of the Bible. This includes the concept of supernatural predictive prophecy. Consequently, they argue that Daniel was written after the facts it seems to predict. Particularly, Daniel 8 explicitly refers to the Medo-Persian and Grecian empires. And the first 20 verses of Daniel 11 are very accurate predictions of the Greek kings. Critical scholars, therefore, argue that these prophecies were written after the fact as history in the form of prophecy. That, however, puts a use question mark over the Book of Revelation, for it depends heavily on Daniel. And is also puts a question mark on the entire Bible for including a piece of fraud.

Many commentators have attempted to defend the historicity of Daniel on historical, linguistic, and other grounds. See, for example - Is Daniel a fraud? But such efforts do not impress the world of science with its strong anti-supernatural bias.

But I would like to add two things that prove that Daniel is true prophecy. Critical scholars believe that Daniel was written in the second century BC, bat Daniel 9 provides a very accurate prediction of Jesus Christ in the first century AD. Furthermore, as another answer has indicated, the fall of Rome in the fifth century AD was also very accurately predicted by Daniel. See - The Fall of Rome shows the Book of Daniel to be true prophecy.

The prophecy of the great image in Dan 2:31-35 is one of the few Bible prophecies about which there is broad agreement, largely because it is explicitly interpreted in Dan 2:37-45. In outline, there are five kingdoms depicted:

1. The head of gold is Babylon, Dan 2:37, 38. This is agreed by Ellicott, Barnes, Jamieson-Fausset-Brown Bible Commentary, Gill, Poole, Cambridge Commentary, Pulpit commentary, Geneva Bible, etc. The Neo-Babylonian empire was founded by Nabopolassar in 626 BC, inherited by Nebuchadnezzar the Great in 605 BC but ended with the capture of Babylon by the Persians in 539 BC.
2. The Chest & arms of silver is Medo-Persia, Dan 2:39. This is agreed by Ellicott, Barnes, Jamieson-Fausset-Brown Bible Commentary, Gill, Poole, Cambridge Commentary, Geneva Bible, etc. The Medo-Persians conquered Babylon in 539 BC, and were themselves overthrown by Alexander the Great in about 330 BC.
3. The Belly of Bronze is Greece, Dan 2:39. This is agreed by Ellicott, Barnes, Jamieson-Fausset-Brown Bible Commentary, Gill, Poole, Cambridge Commentary, Geneva Bible, etc. The Greeks over-ran the great world powers at the time but finally were overtaken by Rom in the middle of the second century BC.
4. The legs of Iron are Rome, Dan 2:40. This is agreed by Ellicott, Bension, Barnes, Jamieson-Fausset-Brown Bible Commentary, Poole, Gill, Geneva Bible, Pulpit Commentary, etc. The Roman empire lasted, united, until the late 5th century AD, whereupon it collapsed and broke up u=into the various independent sates that now surround the Mediterranean sea, namely, Europe, North Africa and the middle east; this is depicted by the feet of ion and clay that will not mix or "cleave" to each other.
5. The Great Rock cut out without hands is the eternal kingdom of Jesus Christ, Dan 2:44, 45. Thus all commentaries agree again.

The place at which most commentators diverge is the point where the Kingdom of Christ is established. There are two schools of thought:

• The Kingdom of Christ was established by Jesus Christ during His first advent (as the Christian church) and will continue and be subsumed by Jesus at His second advent
• The Kingdom of Christ will be established by Jesus at His second advent (Rev 11:15) and will last eternally.

Both views have their ardent supporters and both can be defended. I personally have trouble selecting either as I believe both are correct, but that is a topic for another question.