No, you cannot hold your employer liable for withholding too little taxes. Since the taxes were your responsibility, it would have been your duty to contact them in the beginning of the year when you realized they were not taking anything out.
Social Security and Medicare taxes are standard across the board. There are not special deductions or other considerations. Everyone who is employed and subject to FICA withholdings has 7.65% withheld for SS and Medicare whether you have no kids or 15 kids.
Now if they were not properly withholding your FICA taxes or withheld taxes and did not make the payments to the IRS then they could face penalties, but there are not repercussions for them not withholding enough for you.
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