"Proving" depends on whether the recipient (Bob) cooperates (i.e. accepts to reveal his private key to the verifier), and also on the type of cryptographic algorithms and details of the key.
If Bob cooperates, then he may decrypt the message; this may show that the message "makes sense" when decrypted with Bob's private key, which is a rather strong hint that Bob was intended as the recipient, or at least as a recipient. This depends on what qualifies as "making sense", of course. Most asymmetric encryption systems are hybrid: an asymmetric key exchange yields a message-specific key shared between sender and receiver, and the key is then used to encrypt the actual message data. If the encryption uses (as it should) a MAC to detect alterations, then this can be turned into a convincing proof that the sender really worked with Bob's public key in mind (but it depends on the exact algorithms employed).
Note that, depending on the used algorithms, it can be possible for Bob to take an existing message, and compute his own key pair so that the message (as a sequence of bytes), when decrypted with his brand new private key, decrypts to a message of his choosing. Thus, asymmetric encryption cannot, in general, be considered equivalent to an ownership claim.
If Bob does not cooperate, things can be tricky. We are now in the model of traffic analysis, where an attacker tries to see through anonymous communications. In that model, Bob is not a potential attacker, but a potential victim. Hiding the recipient identity is not a primary property of asymmetric encryption and key exchange systems. If we consider asymmetric encryption with RSA, then the key size is leaked, since it matches the encrypted message size; this can reduce the number of possible recipients.
As a more extreme case, consider Diffie-Hellman. DH is computed in a given group, consisting of a prime modulus p, a subgroup size q (q divides p-1 and is normally prime) and a subgroup generator g (g has order q, which means that gq = 1 mod p). These values (the "group parameters") are part of Bob's public key. The message will first begin by a value B = gb mod p, with a randomly chosen exponent b: that value is a subgroup element. For a given asymmetrically encrypted message, it is easy to look at the header, see the B value, and check whether it matches a given (p,q,g) DH group specification: simply compute Bq mod p; for the right group, it will yield 1, but (with overwhelming probability) not for a distinct group.
So if potential recipients use Diffie-Hellman and each one generated his own group, then this simple test will pinpoint the actual recipient. On the other hand, if several recipients decided to generate their respective key pairs within the same group (which is allowed in DH and does not incur any security weakness), then outsiders will not be able to determine which recipient is the right one: doing so would entail solving the Decisional Diffie-Hellman problem, which is hard (we don't know of any way to solve DDH, in a normal DH group, except through Discrete Logarithm, which is thwarted by large enough p and q).
Also, pinpointing is not proving. Anybody can take a copy of Bob's DH group parameters (they are part of Bob's public key, thus public themselves) and generate his own key pair within the same group. Therefore, being able to recognize Bob's group in a given message does not demonstrate that only Bob could read the message; the message could be meant for somebody else whose key works in the same group.
(Having many keys in the same group is a very common occurrence when using the elliptic curve version of DH, because generating your own curve is hard work and severely limits interoperability.)