The client authentication in SSL is just that: authentication. It convinces the server that the expected client is indeed on the other end of the line. However, the server obtains no proof that way: there is no data which the server can gather and show to a third party (a judge), which would "obviously" convince the judge that the client was indeed involved in the process and sent the request that the server claims that it received.
The relevant concept is non-repudiation. Ultimately, this is a legal concept. However, that concept, depending on the jurisdiction, will try to use the technical properties of the involved communication protocols as a sturdy (or not) foundation. The client authentication in SSL internally uses a digital signature (as part of the CertificateVerify handshake message) but that signature is computed over random challenge data, but not the application data (HTTP requests and responses) that is sent in the SSL tunnel. Indeed, this signature happens before application data is sent in either direction. Thus, technically, that signature does not provide non-repudiation, since it does not cover the actual data.
In other words, what the server may show to a judge can be a convincing proof that the client did send the alleged request only if the judge assumes that the server applied all needed verifications and is fully honest. However, such a case will be brought to the attention of a judge only in the event of some litigation between the client and the server's owner; as such, it would make little sense to arbitrarily assume that the server is honest (it would make the trial outcome quite biased).
Summary: if you consider the SSL certificate-based client authentication as a "signature", then chances are that the resulting legal value will not hold in court, at least not when the server itself (or its owner) is one of the litigating parties. It may work if you are in a context where it can be somehow proven that the server is completely honest and does not try to frame the user; that's not the usual context when considering digital signatures (usually, digital signatures are requested by the server in order to allow the server to sue the client afterwards).