Axitinib in Treating Patients With Melanoma That is Metastatic or Cannot Be Removed by Surgery

Inclusion Criteria: * Histologically or cytologically proven melanoma (including uveal) that is advanced (metastatic) or unresectable * Measurable disease * No more than two prior regimens (0-2) of systemic therapy for metastatic or recurrent disease; therapy (systemic or radiotherapy) administered in the neo-adjuvant or adjuvant setting for previously localized disease is permitted, provided it was completed more than 3 months prior to enrollment; palliative radiotherapy is permitted provided it is completed \>= 2 weeks prior to study therapy initiation and there is at least one measurable lesion outside the radiation field; at least 2 weeks since the end of prior systemic treatment, radiotherapy, or surgical procedure with resolution of all treatment-related toxicity to National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE) version 4.0 grade =\< 1 or back to baseline except for alopecia or hypothyroidism * Eastern Cooperative Oncology Group (ECOG) performance status of =\< 2 * Life expectancy \>= 12 weeks * Absolute neutrophil count (ANC) \>= 1500 cells/mm\^3 * Platelets \>= 75,000 cells/mm\^3 * Hemoglobin \>= 9.0 g/dL * Creatinine =\< 1.5 X upper limit normal (ULN) or calculated creatinine clearance \>= 60 mL/min * Bilirubin =\< 1.5 X ULN * Transaminase =\< 2.5 X ULN (for documented liver metastases, transaminase up to 5 X ULN is permitted) * Random urinary protein/creatinine ratio \< 2 * Have the ability to swallow and retain oral medicati…