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The next generation of galaxy surveys will allow us to test one of the most fundamental assumptions of the standard cosmology, i.e. that gravity is governed by the general theory of relativity (GR). In this paper, we investigate the ability of the Javalambre Physics of the AcceleratingUniverseAstrophysical Survey (J-PAS) to constrainGR and its extensions. Based on the J-PAS information on clustering and gravitational lensing, we perform a Fisher matrix forecast on the effective Newton constant, mu, and the gravitational slip parameter, eta, whose deviations from unity would indicate a breakdown of GR. Similar analysis is also performed for the DESI and Euclid surveys and compared to J-PAS with two configurations providing different areas, namely an initial expectation with 4000 deg(2) and the future best case scenario with 8500 deg(2). We show that J-PAS will be able to measure the parameters mu and eta at a sensitivity of 2-7 per cent, and will provide the best constraints in the interval z = 0.3-0.6, thanks to the large number of ELGs detectable in that redshift range. We also discuss the constraining power of J-PAS for dark energy models with a time-dependent equation-of-state parameter of the type w(a) = w(0) + w(a)(1 - a), obtaining Delta w(0) = 0.058 and Delta w(a) = 0.24 for the absolute errors of the dark energy parameters.© 2020 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society ; We are thankful to our colleagues of J-PAS Theory Working Group for useful discussions and to Ricardo Landim for his comments. MAR and ALM acknowledge support from MINECO (Spain) project FIS2016-78859-P (AEI/FEDER, UE), Red Consolider MultiDark FPA2017-90566-REDC, and UCM pre-doctoral grant. JSA acknowledges support from FAPERJ grant no. E-26/203.024/2017; CNPq grant no. 310790/2014-0 and 400471/2014-0; and the Financiadora de Estudos e Projetos -FINEP grants REF. 1217/13-01.13.0279.00 and REF0859/10-01.10.0663.00. SC acknowledges support from CNPq grant nos 307467/2017-1 and 420641/2018-1. This paper has gone through internal review by the J-PAS collaboration. Funding for the J-PAS Project has been provided by the Governments of Espana and Aragon through the Fondo de Inversion de Teruel, European FEDER funding, the MINECO projects AYA2015-66211-C2-1- P, AYA2015-66211-C2-2, and AYA2012-30789; the Brazilian agencies FINEP, FAPESP, FAPERJ and by the National Observatory of Brazil. ; Peer reviewed