At least 22 migrants - many of them children - have drowned trying to reach Kalymnos and Rhodes from Turkey, in the second major boat tragedy off Greek islands this week.
Greek officials said 19 people died and 138 were rescued near Kalymnos.
Greek Prime Minister Alexis Tsipras said he felt "shame" over "the inability of Europe to deal effectively with this human drama".
Speaking in parliament, the left-wing Syriza leader condemned the "level of debate at a senior level, where one is passing the buck to the other" in the EU.
Thousands have been arriving on Lesbos every day after making the dangerous journey from Turkey.
People traffickers usually pack them onto unseaworthy boats, often inflatable dinghies.