You can always try and apply for an advance parole on "humanintarian" grounds based on your ailing grandma. As far as I am concerned, a "sick grandma" will not cut it, but you can always try and see for yourself. If granted the AP, returning with a US born child is no problem.
Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.
You need to apply for an "advance parole" travel permit for humanitarian reasons. Your child will need a US passport to accompany you.
Mr. Shusterman's (former INS Trial Attorney, 1976-82) response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.
You need to apply for and obtain Advance Parole prior to your travel, otherwise you will not be able to reenter the United States. Your child should be able to travel on his/her American passport without problems.