No, because its gravitational interactions are negligible. The size of the electron is known, its radius is zero, so it would be a naked singularity, if its gravitational interactions were significant-but they're not, so it isn't. It's just a point particle, moving in flat spacetime.
M. A.Markov, “Elementary particles of maximally large masses (Quarks and Maximons),” Soviet Physics (Journal of Experimental and Theoretical Physics), vol. 24, p. 584, 1967.
V. I. Man’ko and M. A. Markov, “Properties of fridmons and the early stage of evolution of the universe,” Theoretical and Mathematical Physics, vol. 17, no. 2, pp. 1060–1063, 1973.
M. A. Markov, “The maximon and minimon in light of a possible formulation of the concept of an ‘elementary particle’,” JETP Letters, vol. 45, pp. 141–144, 1987.
It is worth noting that, clearly, there does not exist "zero size" in nature. The statement "the radius of the electron is 0" can have only mathematical meaning, not physical. Points exist in mathematics, but even there, their existence is axiomatic.