It has recently been shown that a set of the generalized type IIB supergravity equations follows from the requirement of kappa symmetry of the type IIB Green-Schwarz superstring theory defined on an arbitrary background. In this paper, we show that the whole bosonic part of the generalized type II supergravity equations can be reproduced from the T-duality covariant equations of motion of the double field theory by choosing a non-standard solution of the strong constraint. Then, by using the doubled formalism, we show the Weyl invariance of the bosonic string sigma model on a generalized gravity background. According to the dual-coordinate dependence of the dilaton, the Fradkin-Tseytlin term nicely removes the Weyl anomaly. This result seems likely to support that string theories can be consistently defined on arbitrary generalized supergravity backgrounds.