Observations show a clear vertical metallicity gradient in the Galactic bulge, which is often taken as a signature of dissipative processes in the formation of a classical bulge. Various evidence shows, however, that the Milky Way is a barred galaxy with a boxy bulge representing the inner three-dimensional part of the bar. Here we show with a secular evolution N-body model that a boxy bulge formed through bar and buckling instabilities can show vertical metallicity gradients similar to the observed gradient if the initial axisymmetric disk had a comparable radial metallicity gradient. In this framework, the range of metallicities in bulge fields constrains the chemical structure of the Galactic disk at early times before bar formation. Our secular evolution model was previously shown to reproduce inner Galaxy star counts and we show here that it also has cylindrical rotation. We use it to predict a full mean metallicity map across the Galactic bulge from a simple metallicity model for the initial disk. This map shows a general outward gradient on the sky as well as longitudinal perspective asymmetries. We also briefly comment on interpreting metallicity gradient observations in external boxy bulges.