We consider the detection of binary (antipodal) signals transmitted in a spatially multiplexed fashion over a fading multiple-input multiple-output (MIMO) channel and where the detection is done by means of semidefinite relaxation (SDR). The SDR detector is an attractive alternative to maximum likelihood (ML) detection since the complexity is polynomial rather than exponential. Assuming that the channel matrix is drawn with i.i.d. real valued Gaussian entries, we study the receiver diversity and prove that the SDR detector achieves the maximum possible diversity. Thus, the error probability of the receiver tends to zero at the same rate as the optimal maximum likelihood (ML) receiver in the high signal to noise ratio (SNR) limit. This significantly strengthens previous performance guarantees available for the semidefinite relaxation detector. Additionally, it proves that full diversity detection is in certain scenarios also possible when using a non-combinatorial receiver structure.