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 medication * No evidence of preexisting uncontrolled hypertension as documented by 2 baseline blood pressure readings taken at least 1 hour apart; the baseline systolic blood pressure readings must be =\< 140 mm Hg, and the baseline diastolic blood pressure readings must be =\< 90 mm Hg; patients whose hypertension is controlled by antihypertensive therapies are eligible * Women of childbearing potential must have a negative serum or urine pregnancy test within 3 days prior to treatment * Patients of child-bearing potential must agree to use acceptable contraceptive methods (e.g., double barrier) during treatment and for 6 months following completion of study treatment * Patient or legal representative must understand the investigational nature of this study and sign an Institutional Review Board (IRB) approved written informed consent form prior to receiving any study related procedure Exclusion Criteria: * Prior anti-angiogenic therapy * Major surgery \< 4 weeks or radiation therapy \< 2 weeks of starting the study treatment; prior palliative radiotherapy to metastatic lesion(s) is permitted, provided there is at least 1 measurable lesion that has not been irradiated * Significant history of bleeding events (e.g., hemoptysis, grade 3 or grade 4 gross hematuria) within 6 months prior to registration * Presence of serious non-healing wounds, ulcers (including gastro-intestinal) and bone fractures * Gastrointestinal abnormalities including: * Inability to take oral medication * Requirement for intravenous alimentation * Prior surgical procedures affecting absorption including total gastric resection; segmental small bowel or colon resection is permitted * Treatment for active peptic ulcer disease in the past 6 months * Active gastrointestinal bleeding, unrelated to cancer, as evidenced by hematemesis, hematochezia or melena in the past 6 months without evidence of resolution documented by endoscopy or colonoscopy * Malabsorption syndromes * History of gastrointestinal (GI) perforation within prior 12 months * Current use or anticipated need for treatment with drugs that are known potent cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) inhibitors (i.e., grapefruit juice, verapamil, ketoconazole, miconazole, itraconazole, erythromycin, telithromycin, clarithromycin, indinavir, saquinavir, ritonavir, nelfinavir, lopinavir, atazanavir, amprenavir, fosamprenavir and delavirdine) * Current use or anticipated need for treatment with drugs that are known CYP3A4 or cytochrome P450, family 1, subfamily A, polypeptide 2 (CYP1A2) inducers (i.e., carbamazepine, dexamethasone, felbamate, omeprazole, phenobarbital, phenytoin, amobarbital, nevirapine, primidone, rifabutin, rifampin, and St. John's wort) * Requirement of therapeutic anticoagulant therapy with oral vitamin K antagonists; low-dose anticoagulants for maintenance of patency of central venous access devise or prevention of deep venous thrombosis is allowed; therapeutic use of low molecular weight heparin (or similar parenteral drug) for venous-thromboembolic disease is allowed * Active seizure disorder or evidence of untreated brain metastases, spinal cord compression, or carcinomatous meningitis; patients with brain metastases that have been stable for \>= 4 weeks by radiographic documentation following definitive therapy will be permitted provided this is not the only site of metastatic disease * Arterial thrombotic events within 6 months of registration, including myocardial infarction, unstable angina or angina requiring medical or surgical intervention in the past 6 months, coronary/peripheral artery bypass graft, cerebrovascular accident, transient ischemic attack and clinically significant peripheral vascular disease (i.e., claudication on less than 1 block) * Current congestive heart failure (New York Heart Association \[NYHA\] class II, III or IV) * Known human immunodeficiency virus (HIV) or acquired immunodeficiency syndrome (AIDS)-related illness * History of a malignancy except those treated with curative intent for skin cancer (other than melanoma), in-situ breast or in-situ cervical cancer, or those treated with curative intent for any other cancer with no evidence of disease for 3 years * Female patients who are pregnant or lactating * Received an investigational agent within 30 days prior to enrollment * A serious uncontrolled medical disorder or active infection that would impair their ability to receive study treatment * Any condition which in the investigator's opinion would prohibit the understanding or rendering of informed consent and compliance with the requirements of this protocol