It is well known that you can compute any combinatoric function (and hence, any encryption method) with a sufficient number of NAND gates, and so the answer to your question would appear to be "yes, any algorithm is able to run on a "physical level", without a fancy GUI, multitheaded operating system, memory, processor or even a motherboard".
However, I do notice that in the question title, you mention "Encryption", while in the text, you mention "Signing". Now, we in the cryptographical community use those terms to mean rather different things:
Encryption means a method of transforming the data in such a way that someone without the 'key' cannot read it (but someone with the key can).
Signing means a method of producing a tag with the data and a 'private key', in such a way that someone with the 'public key' can verify that the data and the tag correspond to each other.
In addition, there is a third option, which we don't call signing, but is sometimes confused with it:
- A Message Authentication Code (MAC) is a method of producing a tag with the data and a 'key' in such a way that someone else with the 'key' can verify that the data and the tag correspond to each other.
Signatures and MAC do sound similar; the distinction is that with a MAC, the person authenticating the MAC can also generate valid MACs if he wants; with a signature, someone with only the public key cannot generate valid signatures. This is a distinction that sometimes comes in handy, however known signature methods are considerably more expensive than MACs. Because of this, we never use a signature unless we need this additional property.
Now that we have defined the terms, we can get back to your question: is your question about encryption, signature generation or MACs?
Well, in practice, it is quite possible to encrypt data or generate MACs without any processor being involved, and we actually sometimes do it in practice.
When we're talking about generating signatures, well, it's more questionable. Generating a signature is considerably more complex; you could do it without an internal processor (using a pile of NAND gates), however, it's not clear why anyone would choose to. I don't know the internals of every crypto chip that does signatures (or even the internals of everything I've used), but of the ones that I do know, they all have an internal processor involved (at the very least, to orchestrate the signature process)