Sorry no, if you mailed the federal you have to mail state. You can only efile state if federal was first efiled and accepted.
New York prefers e-filed returns, but there are certain circumstances that make it necessary for you to file your NY return by mail and the state DOES allow mailed returns. If you could not or did not e-file your federal return then your NY return cannot be e-filed. It has to be mailed. It is not possible to e-file any state return if the federal return was not e-filed and accepted by the IRS.
https://www.tax.ny.gov/pit/efile/efile_mandate_for_individuals.htm
https://www.tax.ny.gov/pit/file/return_assembly_mail.htm
Unfortunately, Turbotax can't e-file a state return unless the federal return was also e-filed. This is a security measure to prevent scammers from filing multiple state returns, and I believe it applies to all consumer-level tax software.
A professional might be able to e-file a state return, but you would have to pay for that separately.
Some states allow e-filing directly on their web site, but New York is not one, they just point you to consumer software.
You will have to print, sign and mail your return.