The divorce decree passed by the US court is valid one in India, if it is comply the conditions such as :
( I) The relief sought before the US Court shall be on the grounds available in the matrimonial law under which the parties are married;
(ii) The respondent shall voluntarily and effectively submits to the jurisdiction of the US Court and contested proceedings.
(ii) The divorce decree shall be on the merits of the case.
The divorce petition on the grounds of mental cruelty and adultery are available in Hindu Marriages Act. Therefore, the decree of US Court is valid in India, if your wife contest the divorce petition filed by you.
In case, if your wife is not contest the case and if you get an exparte decree, the same is not valid in India. Because, in future, your wife may file a petition in India claiming that she had not availed proper opportunity to contest the case filed you in US Court. The exparte decree is not a valid decree and cannot be executable in India.
Therefore, if your wife is returned to India and staying in India then it is better to you to approach the Jurisdictional Court in India to file the divorce case to avoid executable difficulties.
Hope, i have cleared your doubts. If your need any further clarification, do not hesitate to contact me. Thanks.