1. A method for producing a lithographic printing plate comprising in order:
a step of preparing a lithographic printing plate precursor having an image-recording layer on a hydrophilic support;
a step of exposing the lithographic printing plate precursor in an image shape; and
a step of removing a non-exposed portion of the image-recording layer exposed in the lithographic printing plate precursor using a developer,
wherein the developer includes a compound having at least one acidic group selected from the group consisting of a phosphate group, a phosphonate group, and a phosphinate group and one or more carboxy groups,
a pH of the developer is 5 to 10, and
after the removal step, a step of washing a surface of an obtained lithographic printing plate with water and a step of desensitizing the surface of the obtained lithographic printing plate are not provided.
2. The method for producing a lithographic printing plate according to claim 1,
wherein the developer further includes an anionic surfactant and a nonionic surfactant.
3. The method for producing a lithographic printing plate according to claim 2,
wherein a mass ratio (the nonionic surfactant to the anionic surfactant) between a content of the nonionic surfactant and a content of the anionic surfactant in the developer is 1.2:1.Oto 5.0:1.0.
4. The method for producing a lithographic printing plate according to claim 2 or 3,
wherein a total content of the nonionic surfactant and the anionic surfactant in the developer is 2% by mass to 20% by mass of a total mass of the developer.
5. The method for producing a lithographic printing plate according to any one of claims 1 to
4,
wherein the image-recording layer contains an infrared-absorbing colorant, a polymerization initiator, a polymerizable compound, and a binder polymer.

6. The method for producing a lithographic printing plate according to any one of claims 1 to
4,
wherein the image-recording layer contains an infrared-absorbing colorant, a polymerization initiator, a polymerizable compound, and a particulate polymer compound.
7. The method for producing a lithographic printing plate according to any one of claims 1 to
4,
wherein the image-recording layer contains an infrared-absorbing colorant and a thermoplastic resin particle.
8. The method for producing a lithographic printing plate according to any one of claims 1 to
7,
wherein the number of carboxy groups in the compound having at least one acidic group selected from the group consisting of a phosphate group, a phosphonate group, and a phosphinate group and one or more carboxy groups is 2 or more and 10 or less.
9. The method for producing a lithographic printing plate according to any one of claims 1 to
8,
wherein a content of the compound having at least one acidic group selected from the group consisting of a phosphate group, a phosphonate group, and a phosphinate group and one or more carboxy groups is 0.1% by mass to 10% by mass of the total mass of the developer.
10. The method for producing a lithographic printing plate according to any one of claims 1
to 9,
wherein the developer does not include a water-soluble polymer compound or a content of the water-soluble polymer compound is more than 0% by mass and 0.05% by mass or less of the total mass of the developer.