  A method to construct noncommutative instantons as deformations from
commutative instantons was provided in arXiv:0805.3373. Using this
noncommutative deformed instanton, we investigate the spinor zero modes of the
Dirac operator in a noncommutative instanton background on noncommutative R^4,
and we modify the index of the Dirac operator on the noncommutative space
slightly and show that the number of the zero mode of the Dirac operator is
preserved under the noncommutative deformation. We prove the existence of the
Green's function associated with instantons on noncommutative R^4, as a smooth
deformation of the commutative case. The feature of the zero modes of the Dirac
operator and the Green's function derives noncommutative ADHM equations which
coincide with the ones introduced by Nekrasov and Schwarz. We show a one-to-one
correspondence between the instantons on noncommutative R^4 and ADHM data. An
example of a noncommutative instanton and a spinor zero mode are also given.
