It is shown that non-Abelian string supported in N=2 supersymmetric QCD with U(N) gauge group and |number of quark multiplets 2N can behave as a fundamental superstring. In addition to translational moduli the non-Abelian string carry also orientational/ size moduli. Their dynamics is described by two dimensional weighted CP(N,N) model on the string world sheet. For N=2 the dimension of orientational/ size moduli space is six and can be combined with four translational moduli to form a ten dimensional space required for a superstring to be critical. For this case CP(2,2) model|on the string world sheet is defined on the non-compact Calabi-Yau conifold. We study the closed string states which emerges in four dimensions and identify them with mesons of the bulk theory. We find one massless hypermultiplet associated with the deformation of the complex structure of the conifold.