A Study of Ramucirumab (LY3009806) in Combination With Paclitaxel in Participants With Gastric Cancer

Inclusion Criteria: * The participant has a diagnosis of gastric or GEJ adenocarcinoma. * The participant has disease progression during or within 4 months after last dose of first-line chemotherapy or during or within 6 months after the last dose of neoadjuvant or adjuvant therapy. * The participant received combination chemotherapy, which must include a platinum and/or a fluoropyrimidine and must not include a taxane or antiangiogenic agent. * The disease is evaluable by imaging per Response Evaluation Criteria in Solid Tumors 1.1. * The participant has an Eastern Cooperative Oncology Group performance status of 0 or 1. * The participant has adequate organ function: * Total bilirubin ≤1.5 × the upper limit of normal (ULN) and alanine aminotransferase (ALT) and aspartate aminotransferase (AST) ≤3 × ULN. If the liver has tumor involvement, AST and ALT \<5 × ULN. * Serum creatinine ≤1.5 × ULN or calculated creatinine clearance ≥50 milliliters/minute. * Urinary protein is \<2+. * Absolute neutrophil count ≥1.5 × 10\^9/liter (L), platelets ≥100 × 10\^9/L, and hemoglobin ≥9 grams/deciliter (5.58 millimoles/L). * International normalized ratio ≤1.5 × ULN and partial thromboplastin time ≤5 seconds above ULN. * The participant has an estimated life expectancy of minimum 12 weeks. * The participant has resolution to Grade 1 or less by Common Terminology Criteria for Adverse Events Version 4.0, of all clinically significant toxic effects of previous therapy. * The participant, if male, is sterile or agrees to use a reliable method of birth control. * The participant, if female, is surgically sterile, is postmenopausal, or agrees to use a highly effective method of birth control. * The participant, if female and of child-bearing potential, must have a negative pregnancy test. Exclusion Criteria: * The participant is receiving therapy with any of the following: * Nonsteroidal anti-inflammatory agents. * Other anti-platelet agents; Aspirin use at doses up to 325 milligrams (mg)/day is permitted. * The participant received radiotherapy within 14 days prior to randomization. * The participant received previous chemotherapy with a cumulative dose of \>900 mg per meter squared (mg/m\^2) of epirubicin or \>400 mg/m\^2 of doxorubicin. * The participant has documented brain metastases or leptomeningeal disease. * The participant has a significant bleeding disorder or vasculitis. * The participant experienced any arterial thromboembolic event within 6 months. * The participant has symptomatic congestive heart failure or symptomatic cardiac arrhythmia. * The participant has uncontrolled hypertension, despite antihypertensive intervention. * The participant underwent major surgery within 28 days. * The participant has a history of gastrointestinal perforation or fistula within 6 months. * The participant has a history of inflammatory bowel disease or Crohn's disease requiring medical intervention within 12 months. * The participant has bowel obstruction or history of chronic diarrhea that is considered clinically significant. * The participant has either of the following: * Child-pugh B or C cirrhosis. * The participant has a serious illness or medical condition including: * Human immunodeficiency virus infection. * The participant has a concurrent active malignancy other than the following: * Nonmelanomatous skin cancer. * In situ carcinoma of the cervix or other noninvasive carcinoma or in situ neoplasm. * The participant has a serious nonhealing: (a) wound, (b) peptic ulcer, or (c) bone fracture. * The participant experienced any Grade 3 or 4 venous thromboembolic event that is not adequately treated.