All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
It is not clear to us from the question whether the marriage took place fulfilling the conditions of a valid marriage contract, the most important of which is the consent (and presence) of the guardian and the presence of two witnesses, or that it was just an agreement between that man and that woman to get married. If it was the first case that took place, then that woman became a wife of that man.
In this case, it is not permissible for his first wife to ask him to divorce his second wife; Abu Hurayrah narrated that the Prophet said: “It is not lawful for a woman to ask for the divorce of her sister (i.e. the other wife of her husband) in order to have everything for herself.” [Al-Bukhari]
Therefore, it is not permissible for her (the first wife) to ask for divorce due to his second marriage. Besides, it is foolishness from her to threaten him to commit suicide.
On the other hand, the second wife should try to convince him to keep her and not divorce her, but in case he insists on divorcing her, then she should accept that and submit herself to the Decree of Allaah as this may be good for her and Allaah may bless her instead with another husband who is better than him. Allaah Says (what means): {But if they separate [by divorce], Allaah will enrich each [of them] from His abundance. And ever is Allaah Encompassing and Wise.}[Quran 4:130] She may also seek the help of some trustworthy Muslims in looking for a pious husband for her.
This is of course if a valid marriage contract took place between her and that man. But if the matter was just an agreement between them to get married, then this is a void marriage as it took place without the consent of the guardian and the presence of two witnesses, whereas the consent of the guardian is a condition for the validity of marriage without which marriage is void according to the view of the majority of the scholars. The same applies to the presence of the witnesses, even if some scholars did not condition the presence of the witnesses at the marriage contract, they conditioned it when consummating the marriage or prior to the consummation of the marriage. For more benefit, please refer to Fatwa 83629.
Finally, we advise that sister to protect her religion and be steadfast on it even if that husband separated from her. Indeed, a Muslim is tested according to the strength of his/her religion. The Prophet said: “When Allaah loves a people, He tries them; whoever is satisfied will get satisfaction, and whoever is discontented will get displeasure.” [At-Tirmithi and Ibn Maajah] The Prophet also said: “How strange is the matter of a believer, there is good in all his affairs, and this is only for the believer. If something good happens to him and he is grateful, then this is good for him, and if a calamity befalls him and he is patient, then this is also good for him.” [Muslim] For more benefit, please refer to Fatwa 83577.