We develop a unified framework for testing gravity beyond general relativity (GR) with continuous gravitational waves (CWs) from individual supermassive black hole binaries. These long-lived, nearly monochromatic nanohertz signals offer unique strengths for precision tests of gravity, since their coherent phase evolution and inter-pulsar correlations in pulsar timing arrays (PTAs) retain detailed information about departures from GR over cosmological propagation distances. We consider three representative classes of deviations from GR: additional polarization states, modified dispersion relations, and parity-violating birefringence. For each, we derive the interpulsar cross-correlation, the modified antenna response, and the propagation-induced pulsar-term phase delay. For nontensorial polarizations, the CW cross-correlation scales linearly in the alternative-polarization amplitude, compared to the quadratic scaling of the gravitational-wave background, provided the beyond-GR modes are subdominant. PTAs are also competitive for modified dispersion relations, where low frequencies enhance both the antenna-pattern modification and the pulsar-term phase delay. Birefringence, by contrast, is suppressed at nanohertz frequencies for most parity-violating theories. We validate the framework with injection-and-recovery simulations for breathing-mode and massive-graviton signals at current observational limits, recovering the injected beyond-GR parameters and distinguishing the CW signal from both correlated and uncorrelated background models. We further show that a pure-GR CW template recovers source parameters without significant bias when beyond-GR physics is present in the data, supporting a two-stage analysis strategy: identify candidates under GR, then test for deviations.

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• Alternative gravity theories
• General relativity
• Gravitational waves