A new Galerkin-type meshfree method is developed for solving incompressible Navier-Stokes equations by integrating the key strengths of the semi-Lagrangian (SL) method and the element-free Galerkin (EFG) method. This integration not only effectively resolves the convection-dominance problem but also fully preserves the meshfree property of the EFG method. In the absence of grid constraints, the operations of backward tracing and interpolation in the SL method can be executed more conveniently. To achieve both good stability and accuracy, the SL method is employed to handle the convection terms, while the EFG method is utilized for the diffusion terms. To decouple the velocity and pressure, a novel fractional step algorithm is derived within the SL framework. This algorithm circumvents the Ladyzhenskaya-Babuška-Brezzi (LBB) constraint and permits the utilization of equal-order velocity-pressure interpolation. Given that the SL method exhibits unconditionally stable characteristics for convection terms, the Galerkin method offers an optimal approximation for diffusion terms, the fractional step algorithm decouples velocity and pressure variables, and the meshfree feature streamlines the implementation of the SL method, the proposed method is anticipated to be an efficient approach for solving the incompressible Navier-Stokes equations. Numerical examples with available analytical solutions are solved to show the accuracy, stability and convergence behavior of the proposed method. The results demonstrate that the new method exhibits superior stability compared to the EFG method, and it reaches to a first-order convergence rate in temporal direction and second-order convergence rate in spatial direction under first-order discretization. After that, numerical tests on the square-cavity driven flow and the doubly periodic shear layer flow further validate the accuracy and stability of the proposed method.