I do not know the laws for India. I am originally from Canada and we might have a good analogy from those tax rules. My brother lives in CA and works in the US. He pays income taxes to the US which is credited against the taxes he owes Canada. He does file in both countries but does not pay taxes twice for the same income. If that makes sense you might look into the tax agreement between India and the US.
You might consider looking into a tax ID (ITIN) number for your wife if she does not qualify for a SSN. An ITIN, or Individual Taxpayer Identification Number, is a tax processing number only available for certain nonresident and resident aliens, their spouses, and dependents who cannot get a Social Security Number (SSN). It is a 9-digit number, beginning with the number "9", formatted like an SSN (NNN-NN-NNNN).
To obtain an ITIN, you must complete IRS Form W-7, IRS Application for Individual Taxpayer Identification Number (PDF) . The Form W-7 requires documentation substantiating foreign/alien status and true identity for each individual. You may either mail the documentation, along with the Form W-7, to the address shown in the Form W-7 Instructions, present it at IRS walk-in offices, or process your application through an Acceptance Agent authorized by the IRS. A link is attached to form W-7.
Acceptance Agents are entities (colleges, financial institutions, accounting firms, etc.) who are authorized by the IRS to assist applicants in obtaining ITINs. They review the applicant's documentation and forward the completed Form W-7 to IRS for processing.
NOTE: You cannot claim the earned income credit using an ITIN.
Foreign persons who are individuals should apply for a social security number (SSN, if permitted) on Form SS-5 with the Social Security Administration, or should apply for an Individual Taxpayer Identification Number (ITIN) on Form W-7. Effective immediately, each ITIN applicant must now apply using the revised Form W-7, Application for IRS Individual Taxpayer Identification Number; and attach a federal income tax return to the Form W-7.
Applicants who meet an exception to the requirement to file a tax return (see the instructions for Form W-7) must provide documentation to support the exception.
New W-7/ITIN rules were issued on December 17, 2003. For a summary of those rules, please see the new Form W-7 and its instructions.