Study of IMC-1121B (Ramucirumab) With Best Supportive Care in Participants With Gastric Cancer and Adenocarcinoma

Inclusion Criteria: * Histologically or cytologically confirmed gastric carcinoma, including gastric adenocarcinoma or GEJ adenocarcinoma * Metastatic disease or locally recurrent, unresectable disease with measurable lymph node metastases * Measurable disease and/or evaluable disease. Measurable disease is defined as at least one unidimensionally-measurable target lesion \[≥ 2 centimeter (cm) with conventional techniques or ≥ 1 cm by spiral computed tomography (CT)\], as defined by Response using Response Evaluation Criteria in Solid Tumors (RECIST). Examples of evaluable, nonmeasurable disease include gastric, peritoneal, or mesenteric thickening in areas of known disease, or peritoneal nodules that are too small to be considered measurable by RECIST * Experienced disease progression during or within 4 months after the last dose of first-line therapy for metastatic disease, or during or within 6 months after the last dose of adjuvant therapy * Disease is not amenable to potentially curative resection * Participant is ≥ 18 years of age * Participant has a life expectancy of ≥ 12 weeks * Participant resolution to Grade ≤ 1 (or to Grade ≤ 2 in the case of neuropathy) by the National Cancer Institute Common Terminology Criteria for Adverse Events (NCI-CTCAE), Version 3.0, of all clinically significant toxic effects of prior chemotherapy, surgery, radiotherapy, or hormonal therapy (with the exception of alopecia) * Eastern Cooperative Oncology Group Performance Status (ECOG-PS) score of 0-1 * The participant has adequate hepatic function as defined by a total bilirubin ≤ 1.5 milligrams/deciliter (mg/dL) \[25.65 micromole/liter (µmol/L)\], and aspartate transaminase (AST) and alanine transaminase (ALT) ≤ 3.0 x the upper limit of normal (ULN) \[or 5.0 x the ULN in the setting of liver metastases\] * The participant has adequate renal function as defined by a serum creatinine ≤ 1.5 x the ULN, or creatinine clearance (measured via 24-hour urine collection) ≥ 40 milliliters/minute (mL/min) (that is, if serum creatinine is \> 1.5 x the ULN, a 24-hour urine collection to calculate creatinine clearance must be performed) * The participant's urinary protein is ≤ 1+ on dipstick or routine urinalysis (\[UA\]; if urine dipstick or routine analysis is ≥ 2+, a 24-hour urine collection for protein must demonstrate \< 1000 milligrams (mg) of protein in 24 hours to allow participation in the study) * The participant has adequate hematologic function, as evidenced by an absolute neutrophil count (ANC) ≥ 1000 microliters (µL), hemoglobin ≥ 9 grams/deciliter (g/dL) \[5.58 millimoles/liter (mmol/L)\], and platelets ≥ 100,000/µL * The participant must have adequate coagulation function as defined by International Normalized Ratio (INR) ≤ 1.5 and a partial thromboplastin time (PTT) ≤ 5 seconds above the ULN (unless receiving anticoagulation therapy). Participant on anticoagulation therapy with unresected primary tumors or local tumor recurrence following resection are not eligible * If the participant has received prior anthracycline therapy as part of his or her first-line regimen, the participant is able to engage in ordinary physical activity without significant fatigue or dyspnea * Because the teratogenicity of IMC-1121B is not known, the participant, if sexually active, must be postmenopausal, surgically sterile, or using effective contraception (hormonal or barrier methods) * Female participant of childbearing potential must have a negative serum pregnancy test within 7 days prior to randomization * Able to provide informed written consent and is amenable to compliance with protocol schedules and testing Exclusion Criteria: * Documented and/or symptomatic brain or leptomeningeal metastases * Experienced any Grade 3-4 gastrointestinal bleeding within 3 months prior to randomization * Experienced any arterial thromboembolic events, including but not limited to myocardial infarction, transient ischemic attack, cerebrovascular accident, or unstable angina, within 6 months prior to randomization * Ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, symptomatic or poorly controlled cardiac arrhythmia, uncontrolled thrombotic or hemorrhagic disorder, or any other serious uncontrolled medical disorders in the opinion of the investigator * Ongoing or active psychiatric illness or social situation that would limit compliance with study requirements * Uncontrolled or poorly-controlled hypertension despite standard medical management * Participant has a serious or nonhealing wound, ulcer, or bone fracture * Received chemotherapy, radiotherapy, immunotherapy, or targeted therapy for gastric cancer within 2 weeks prior to randomization * Received any investigational therapy within 30 days prior to randomization * Undergone major surgery within 28 days prior to randomization, or subcutaneous venous access device placement within 7 days prior to randomization * Received prior therapy with an agent that directly inhibits vascular endothelial growth factor (VEGF) or VEGF receptor 2 (R-2) activity (including bevacizumab), or any antiangiogenic agent * Receiving chronic antiplatelet therapy, including aspirin, nonsteroidal anti-inflammatory drugs (NSAIDs, including ibuprofen, naproxen, and others), dipyridamole or clopidogrel, or similar agents. Once-daily aspirin use \[maximum dose 325 milligram/day (mg/day)\] is permitted * Participant has elective or planned major surgery to be performed during the course of the clinical trial * Participant has a known allergy to any of the treatment components * Pregnant or lactating * Known to be positive for infection with the human immunodeficiency virus * Known alcohol or drug dependency * Participant has a concurrent active malignancy other than adequately-treated nonmelanomatous skin cancer, other noninvasive carcinoma, or in situ neoplasm. A participant with previous history of malignancy is eligible, provided that he/she has been free of disease for \> 3 years