Praise be to Allah.
Man-woman relationship before marriage
Before responding to this question, a comment must be made regarding an objectionable and reprehensible issue implied in the question, and that is the mentioning by the sister that he “falls in love with another woman”.
It is known that it is not permissible in the Islamic religion to establish a relationship between a man and a woman who is ajanabiyya to him (lit. foreign, i.e., marriageable) before marriage, for Allah revealed in Surat al-Maidah (the Table), ayah 5 a verse whose meaning can be translated as:
“This day are (all) things good and pure made lawful unto you. The food of the People of the Book is lawful unto you and yours is lawful unto them. (Lawful unto you in marriage) are (not only) chaste and virtuous women who are believers, but chaste women among People of the Book, revealed before your time, when you give them their due dowers, and desire chastity, not lewdness, nor secret intrigues. If anyone rejects faith, fruitless is his work, and in the Hereafter he will be in the ranks of those who have lost (all spiritual good).”
And the word “akhdan” that appears in this ayah (appearing above as “secret intrigues”) means intimate friend or companion, and in this case refers to a lover , and Allah has indicated in Surat al-Ahzab (33:53) that a condition for talking with women when it is needed is: (a translation of the meaning)
“…and if you ask them (women) for something you want, ask them from behind a hijab (both in the sense of a physical barrier such as a screen or wall, as well as in clothing); that makes for greater purity for your hearts and for theirs…”
And Allah has commanded women not to speak with soft, sensual voices with a man who is ajnabi (lit. foreign, meaning one to whom marriage is not forbidden) so that he is not tempted by their voices, and so as not to provoke feelings of lust. Allah ta'ala said in Surat al-Ahzab, ayah 32, a verse whose meaning can be translated as:
“… if you do fear (Allah) be not too complacent of speech lest one in whose heart is a disease should be moved with desire; but utter sufficient and appropriate speech.” (33:32)
So how could it possibly be permissible after all of these injunctions to establish love or friendship relations among women and men who are ajanib (marriageable)?
Conditions to marry a second wife in Islam
And as for the original matter of the question, Allah (subhanahu wa-ta’ala), Who is al-Hakim (All Wise) al-Khabir (All Experienced), and Who is more knowledgeable about the human beings He created than they themselves are about themselves, has ordained that a man is allowed to marry whom he wills among women as long as he does not bond in marriage under his care and responsibility more than four women .
And this is conditioned upon him dealing justly among them the known justice specified by Shari’ah which includes overnight stays and spending.
If he does not have the ability and capacity to deal with such justice , then he should suffice with one, as Allah indicates in Surat al-Nisa in ayah 3, which has a meaning that can be translated as:
“… marry women of your choice, two, or three, or four; but if you fear that you will not be able to deal justly (with them) then only one…” (4:3)
And Allah’s Shari’ah (law) is all just and wise, and He ordains what He wills and does what He wants and it is incumbent upon as human beings to believe and have faith and submit and obey and abide by the Shari`ah, otherwise then we are not Muslims nor mu’minin (believers). As Allah also said in Surat al-Nisa, ayah 65, a verse whose meaning can be translated as:
“But no, by your Lord, they can have no (real) faith until they make you judge in all disputes between them, and find in their souls no resistance against your decisions, but accept them with the fullest conviction” (4:65)
and in Surat al-Ahzab, ayah 36, a meaning that can be translated as:
“It is not fitting for a mu’min (believing man) or mu’minah (believing woman) when a matter has been decided by Allah and His Messenger to have any option about their decision: if anyone disobeys Allah and His Apostle he is indeed strayed on a manifestly wrong path.” (33:36)
Second marriage in Islam: Do you need permission from your first wife?
Furthermore, no evidence appears either in the Quran or sunnah requiring the permission of the first wife if her husband wishes to marry another wife, and therefore he is not required to ask her permission.
However, he needs to be judicious in taking this decision and to weigh it carefully with respect to benefits and drawbacks and to look with the eye of wisdom at all of the considerations pertaining to the matter, and he should strive all he can to conciliate, reassure, and satisfy his first wife, in order to ease and mitigate the effect of the matter upon her.