Where in the Bible is the restoration of the nation of Israel prophesied?

Your question can be answered in two ways:

1. Chapter and verse to a specific prophecy that clearly has a 1:1 correspondence to the nation of Israel today.

2. The sum of biblical teaching and how it evolves from OT prophecies and NT affirmative and clarifying statements.

In a matter like this one I think the 2nd option clearly is the best one. I would build my case along these lines:

• The promises given to Abraham and his descendants over the line of Isaac and Jacob do retain a geographic significance, even if there are many spiritual and typological/allegorical interpretations that will provide further knowledge. God does not back-peddle on his promises (Rom 11:29). Thus, the church fathers in the 2nd century and later, that substituted the literal meaning for allegorical meanings, were (IMO) wrong, thanks to their general anti-judaism.
• The promises of restored nationhood given by prophets to Israel, while in exile, where not completely fulfilled upon their immediate return, nor were they completely fulfilled in the Maccabean era. This is evidenced by the question to Jesus in Acts 1:6-8, when Jesus is asked post-resurrection, if the time has come to restore Israel. Jesus dismisses the question while not invalidating its basic premise. It is not for the disciples to know the time and manner, but it will happen.
• Thus, any eschatological scheme must be viewed with great suspicion! A thought corroborated by other words of Jesus. And by the enormous amount of failed claims throughout church history. This is my basis for being weary of making 1:1 claims between specific historic events and single pericopes in Scripture.

It would be impossible to provide all details of how my thinking has evolved on this issue, but the end result is that I view present day Israel as the probable fulfillment of biblical prophecies in an embryonic form. I do not think that such belief warrants political support for repressing the Arab population, e.g. as in advocating forced transfer from the west Bank. Ethics must be a consideration, as well as eschatology.

However, if events occurred that would prove today's nation of Israel as just another Maccabean-like era, my faith in God would not be severely shaken, although any such event would surely be the result of a disastrous war or collapse, that would probably ensure a 2nd holocaust. Thus, it should be avoided, anyway, for simple ethical reasons. That means I support the basic tenant of Zionism - that it is a good idea for the Jewish people to have a national home in their ancient homeland - even if its present day manifestation is not necessarily the product of biblical prophecies.

No it isn't a fulfillment of prophesy.The prophesies have already been fulfilled. Gen 13:15 is interpreted by Gal 3:16 Now to Abraham and his seed were the promises made. He saith not, And to seeds, as of many; but as of one, And to thy seed, which is Christ.

2Co 1:19-20 For the Son of God, Jesus Christ, who was preached among you by us, even by me and Silvanus and Timotheus, was not yea and nay, but in him was yea. For all the promises of God in him are yea, and in him Amen, unto the glory of God by us.

Ezek 36 especially 24-27For I will take you from among the heathen, and gather you out of all countries, and will bring you into your own land. Then will I sprinkle clean water upon you, and ye shall be clean: from all your filthiness, and from all your idols, will I cleanse you. A new heart also will I give you, and a new spirit will I put within you: and I will take away the stony heart out of your flesh, and I will give you an heart of flesh. And I will put my spirit within you, and cause you to walk in my statutes, and ye shall keep my judgments, and do them. is fulfilled in John 3:3-8 Jesus answered and said unto him, Verily, verily, I say unto thee, Except a man be born again, he cannot see the kingdom of God. Nicodemus saith unto him, How can a man be born when he is old? can he enter the second time into his mother's womb, and be born? Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God. That which is born of the flesh is flesh; and that which is born of the Spirit is spirit. Marvel not that I said unto thee, Ye must be born again. The wind bloweth where it listeth, and thou hearest the sound thereof, but canst not tell whence it cometh, and whither it goeth: so is every one that is born of the Spirit. The borders of Israel have been reconceived as shown by Jesus's ironic answer to Acts 1:6 in Acts 1:8 But ye shall receive power, after that the Holy Ghost is come upon you: and ye shall be witnesses unto me both in Jerusalem, and in all Judaea, and in Samaria, and unto the uttermost part of the earth.

The restoration of Israel in prophecy has long been fulfilled. Under Ezras leadership the Israelites were taken back to Jerusalem out of exile  Now this is the copy of the letter that King Artaxerxes gave unto Ezra the priest, the scribe, even a scribe of the words of the commandments of the Lord and of His statutes to Israel: 

“Artaxerxes, king of kings, unto Ezra the priest, a scribe of the law of the God of heaven. Perfect peace, and at such a time. I make a decree that all those of the people of Israel and of his priests and Levites in my realm, who are minded of their own free will to go up to Jerusalem, go with thee. - Ezra 7:11-13

This happened in 547 bc

There are two related but distinct biblical issues here. The first is whether the land belongs permanently and unconditionally to the Jews. The second is whether the creation of the modern state of Israel fulfils a prophecy. One does not necessarily follow from the other, since it need not have been God's intention that a Jewish state be set up in the Middle East in 1948.

Genesis 17:8, with its promise to Abraham, appears to support the contention of Israel being forever the possession of the Jews, but it is arguable whether this passage is relevant to the current situation:

And I will give unto thee, and to thy seed after thee, the land wherein thou art a stranger, all the land of Canaan, for an everlasting possession; and I will be their God.

Abraham's first-born was called Ishmael, and he is traditionally considered to be the ancestor of the Arabs:

Genesis 16:15-16: And Hagar bare Abram a son: and Abram called his son's name, which Hagar bare, Ishmael. And Abram was fourscore and six years old, when Hagar bare Ishmael to Abram.

On a biblical basis, this would give the Arabs a strong claim to the land of the Canaanites, but a later passage clarifies this, ensuring that the covenant accrues to the Israelites only.

This article claims the land never unconditionally belonged to Israel, but to God and is at his disposal. Presumably, since Jesus came to fulfil the law, Christian Palestinians also have a strong New Testament claim to the land.

Many evangelical Protestants see a connection between Israel and the fulfilment of biblical prophecy and/or believe that God gave Israel to the Jews in 1948. What no one seems to have been able to identify is exactly which prophecy or prophecies point clearly to the current state of Israel.

If the covenants with Abraham, Isaac and Jacob were permanent and unconditional, the Bible does not explain why the Jews were strangely absent from the land for nearly two thousand years. If it was God's will that the Jews did not occupy the land for so long, it could well be that it is still God's will that they remain abroad. In the absence of a direct link between a prophecy and the current state of Israel, we can not be sure that God intended a Jewish state to be set up at this time.

Ezekiel chapter 36, in which God says that the mountains of Israel will grow branches and bear fruit for his people Israel, who will soon return, is not a prophecy of modern Israel. It is a prophecy of the return of all the Israelites scattered by the Assyrian conquest in 722 BCE and, when read carefully, can only have that meaning. Ezekiel several times prophesied the imminent return of all the Israelites scattered by the Assyrian conquest, but this never happened and, after intermarriage and assimilation, can never happen.

There is no explicit reference to the establishment of modern Israel in the Bible.

In Ezekiel 37, the prophet has a vision of a valley of "dry bones." When asked if these bones could yet live, Ezekiel wisely says, "You alone know, O Lord." (v3) After this, the Lord miraculously makes the bones come together and come alive.

It is explicitly stated in that chapter, that this refers to the people of Israel being re-established in the land of Israel:

Then he said to me: “Son of man, these bones are the people of Israel. They say, ‘Our bones are dried up and our hope is gone; we are cut off.’ Therefore prophesy and say to them: ‘This is what the Sovereign Lord says: My people, I am going to open your graves and bring you up from them; I will bring you back to the land of Israel. - Ezekiel 37:11-12 NIV

Additional context is set in the previous chapter where Ezekiel, speaking for the Lord says:

“‘But you, mountains of Israel, will produce branches and fruit for my people Israel, for they will soon come home. 9 I am concerned for you and will look on you with favor; you will be plowed and sown, 10 and I will cause many people to live on you—yes, all of Israel. The towns will be inhabited and the ruins rebuilt. 11 I will increase the number of people and animals living on you, and they will be fruitful and become numerous. I will settle people on you as in the past and will make you prosper more than before. Then you will know that I am the Lord. 12 I will cause people, my people Israel, to live on you. They will possess you, and you will be their inheritance; you will never again deprive them of their children. - Ezekiel 36:8-12 NIV

Hal Lindsey, (The Late Great Planet Earth), for instance, explains how he sees this as the restoration of Israel in this video Note: Hal Lindsey is a controversial figure, to say the least, but he is the direct inspiration for many premilennial dispensationalists, such as Tim LaHaye (Left Behind), John Walverood, Ed Whisenant, and others.

Ed Whisenant's well-known work 88 Reasons why the Rapture will occur in 1988 was fundamentally based on the conflation of this event, Jesus' parable of the fig tree (not a generation shall pass away until these things occur) and the restoration of Israel in 1948.