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Unfortunately, I'm not sure that this is going to be possible using Apex. I created a test Java program (which works against the Google API) to observe the differences between what Java produces vs Apex. I noted the signatures generated from the two were different which narrowed it down to the output of the Crypto.sign() method.

I found this link which gives the following info:

The Apex Crypto class provides support for Digital Signatures with the sign() method. The following considerations apply:

• The two algorithms are RSA and RSA-SHA1, which are functionally equivalent.
• A PKCS8 formatted private key in base64 decoded form is required. This private key should not be hardcoded in the Apex script but should be stored in a protected custom setting or a encrypted fields in a custom table.
• It is equivalent to the Java Signature.sign() class method using "SHA1withRSA".
• In C#, it is the equivalent of (1) signing the clear text using SHA1Managed.ComputeHash() and (2) Signing using RSACryptoServiceProvider.ComputeHash() against the resulting hash.
• Functionally, it will compute a SHA1 digest from clear text and encrypt the digest using RSA with the provided private key.

I have highlighted the key issue here, I believe you need the equivalent of SHA256withRSA which does not seem to be an option with the Crypto class (at least not that I can figure out).

So, in summary I think your code is correct but the signature being generated is not.