If a woman decides to leave the city of her ex-husband, who gets custody of their child?
Question

                Does the islam permit to use the US courts and laws to solve custody issues? (please assume that both man and woman lives in USA). If not, how can this issue resolove when both parties believe of their custody rights?

If the woman got married, does she loose automaically her custody right to the man?.
Praise be to Allah.With regard to custody, if the former spouses 
    came from different cities, the basic principle concerning custody – so long as the child has not yet reached the age of seven – is that the 
    mother has custody, because the Prophet  (peace and blessings of Allaah be upon him) said: “You have more right to it so long as you have not 
    remarried.” Narrated by Abu Dawood, 1938. This is the basic principle. 
But there is another principle, which is that 
    the matter of custody is based on whatever is in the best interests of the child. If travelling with his mother or father will be harmful to the 
    child, then custody should be given to the parent who will not harm the child by travelling. If the mother’s travelling to another city will not 
    have any harmful effects on the child, then the basic principle is that custody is hers. 
With regard to referring to the courts of a 
    non-Muslim country to resolve the matter of custody, that is not permissible, because this is referring to taaghoot (falsehood, false 
    judges) for judgement, and Allaah says (interpretation of the meaning): 
“And whosoever does not judge by what 
    Allaah has revealed, such are the Kaafiroon (i.e. disbelievers — of a lesser degree as they do not act on Allaah’s Laws)”
[al-Maa'idah 5:44] 
“So judge among them by what Allaah has 
    revealed, and follow not their vain desires, diverging away from the truth that has come to you. To each among you, We have prescribed a law and a 
    clear way”
[al-Maa'idah 5:48]
The two former spouses should go to the 
    Islamic center and ask the scholars or knowledgeable people there to judge between them. 
If the woman remarries, she loses the right to custody, because of the hadeeth quoted above. 
    See also question no. 9463.