The generalized degrees of freedom (GDoF) region of the multiple-input multiple-output (MIMO) Gaussian Z-interference channel with an arbitrary number of antennas at each node is established under the assumption of delayed channel state information at transmitters (CSIT). The GDoF region is parameterized by $α$, which links the interference-to-noise ratio (INR) to the signal-to-noise ratio (SNR) via $INR=SNR^α$. A new outer bound for the GDoF region is established by maximizing a bound on the weighted sum-rate of the two users, which in turn is obtained by using a combination of genie-aided side-information and an extremal inequality. The maximum weighted sum-rate in the high SNR regime is shown to occur when the transmission covariance matrix of the interfering transmitter has full rank. An achievability scheme based on block-Markov encoding and backward decoding is developed which uses interference quantization and digital multicasting to take advantage of the channel statistics of the cross-link, and the scheme is separately shown to be GDoF-optimal in both the weak ($α\leq1$) and strong ($α>1$) interference regimes. This is the first complete characterization of the GDoF region of any interference network with delayed CSIT, as well as the first such GDoF characterization of a MIMO network with delayed CSIT and arbitrary number of antennas at each node. For all antenna tuples, the GDoF region is shown to be equal to or larger than the degrees of freedom (DoF) region over the entire range of $α$, which leads to a V-shaped maximum sum-GDoF as a function of $α$, with the minimum occurring at $α=1$. The delayed CSIT GDoF region and the sum-DoF are compared with their counterparts under perfect CSIT, thereby characterizing all antenna tuples and ranges of $α$ for which delayed CSIT is sufficient to achieve the perfect CSIT GDoF region or sum-DoF.