You can use OpenLayers.Map.updateSize method to redraw your base Layer when you maximize the map.
So, adding a function like:
var fullScreen = function () {
map.baseLayer().redraw();
}
you would only need to add an additional option to your map:
var map = new OpenLayers.Map({
div: "map",
center: new OpenLayers.LonLat(0, 0),
updateSize: fullScreen,
minResolution: "auto",
minExtent: new OpenLayers.Bounds(-1, -1, 1, 1),
maxResolution: "auto",
maxExtent: new OpenLayers.Bounds(-180, -90, 180, 90),
projection: Mercator,
displayProjection: Geographic
});
Good luck!