A foreign Muslim who let his visa expire has now tried to rectify his situation by marrying a non-Muslim with the intention of getting his green card. He paid the Mahr and told his wife that it was a gift and it is hers and not to be taken back. His wife entered the marriage with the intention of being his wife and to help him with any legal problems he may have. On the night of their wedding he did not have sexual relations with her nor has he to the present time. Now he [the husband] wants to divorce her and wants half the Mahr back under the verse 2:237. His intention was clearly to have a temporary marriage to serve his own personal goals. Her intention was to marry a Muslim hoping to receive some benefit to understand Islam and to help her Muslim husband. What would be a just ruling that would give ease and justice to this situation?
They were married in a Masjid in the USA and are married in accordance with US State Law. [They have a legal US marriage].

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Your question consists of some points.

First, you said that this man wanted to marry a non-Muslim woman and you did not specify her religion.  In fact, if this woman is from the People of the Book (Christian or Jew) it is lawful to marry her provided she is chaste.  Allah Says (interpretation of meaning): {(….Lawful to you in marriage) are chaste women from the believers and chaste women from those who were given the Scripture (Jews and Christians) before your time, when you have given their due Mahr (bridal money given by the husband to his wife at the time of marriage), desiring chastity (i.e. taking them in legal wedlock) not committing illegal sexual intercourse, nor taking them as girl-friends…..}[5:5].

But if this woman is not from the People of the Book, then it is unlawful for a Muslim to marry her, until she becomes Muslim.  Allah Says (interpretation of meaning): {And do not marry Al-Mushrikât (idolatresses, etc.) till they believe (worship Allâh Alone)…..}[2:221].

Second, if the man had told this woman before marrying that his aim was to get official papers, then this condition renders this marriage a temporary marriage, hence unlawful.  The temporary marriage means that the man and the woman agree to remain married for a given period or until something happens.

Now, if the man did not specify such a condition but only had that intention in mind, then this is lawful.

Third, his saying that the Mahr is a gift, then the woman owned it legally as soon as she possessed it, and in this case, he has to give another Mahr.  But if his intention was that the gift was a Mahr and he wants to take back half of it because he did not have sex with his wife, then this is a matter where there is disagreement among the scholars.

The majority of scholars are of the opinion that staying in seclusion with one’s wife after the contract gives her full ownership of the Mahr even if the man did not have sex with her.

The evidence for this is the authentic narrations by Imam Ahmad from Zurarah who said: 'The four Caliphs applied it that whenever the man draws up curtains or closes the door hence remaining in seclusion with his wife, this entitles her for the Full Mahr, and if he divorces her she has to fulfill Iddah'.

It is understood from your question that she stayed with him the first night.  So, he has no right to take back any portion of the Mahr.

On the other hand, it would have been much more appropriate for this man not to break his relation with this woman until she becomes Muslim hence saving her from Hell-fire.

Indeed, convincing someone to become Muslim is highly rewarded by Allah.

Allah knows best.