According to Java 7 documentation as well as third party vendors, it appears Java 7 should support AES-GCM suites:
There are no GCM cipher suites in the SunJSSE Provider of Java 7 (assuming an Oracle JRE), although it supports TLS 1.2.
These have been introduced in Java 8 (see cipher suite table in the "The SunJSSE Provider" section).
1.8.0-ea-b124
Connecting with 71 cipher suites supported:
********* TLS_ECDHE_ECDSA_WITH_AES_128_CBC_SHA256 *********
********* TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA256 *********
********* TLS_RSA_WITH_AES_128_CBC_SHA256 *********
********* TLS_ECDH_ECDSA_WITH_AES_128_CBC_SHA256 *********
********* TLS_ECDH_RSA_WITH_AES_128_CBC_SHA256 *********
********* TLS_DHE_RSA_WITH_AES_128_CBC_SHA256 *********
********* TLS_DHE_DSS_WITH_AES_128_CBC_SHA256 *********
********* TLS_ECDHE_ECDSA_WITH_AES_128_CBC_SHA *********
********* TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA *********
********* TLS_RSA_WITH_AES_128_CBC_SHA *********
********* TLS_ECDH_ECDSA_WITH_AES_128_CBC_SHA *********
********* TLS_ECDH_RSA_WITH_AES_128_CBC_SHA *********
********* TLS_DHE_RSA_WITH_AES_128_CBC_SHA *********
********* TLS_DHE_DSS_WITH_AES_128_CBC_SHA *********
********* TLS_ECDHE_ECDSA_WITH_RC4_128_SHA *********
********* TLS_ECDHE_RSA_WITH_RC4_128_SHA *********
********* SSL_RSA_WITH_RC4_128_SHA *********
********* TLS_ECDH_ECDSA_WITH_RC4_128_SHA *********
********* TLS_ECDH_RSA_WITH_RC4_128_SHA *********
********* TLS_ECDHE_ECDSA_WITH_AES_128_GCM_SHA256 *********
********* TLS_ECDHE_RSA_WITH_AES_128_GCM_SHA256 *********
********* TLS_RSA_WITH_AES_128_GCM_SHA256 *********
********* TLS_ECDH_ECDSA_WITH_AES_128_GCM_SHA256 *********
********* TLS_ECDH_RSA_WITH_AES_128_GCM_SHA256 *********
********* TLS_DHE_RSA_WITH_AES_128_GCM_SHA256 *********
********* TLS_DHE_DSS_WITH_AES_128_GCM_SHA256 *********
********* TLS_ECDHE_ECDSA_WITH_3DES_EDE_CBC_SHA *********
********* TLS_ECDHE_RSA_WITH_3DES_EDE_CBC_SHA *********
********* SSL_RSA_WITH_3DES_EDE_CBC_SHA *********
********* TLS_ECDH_ECDSA_WITH_3DES_EDE_CBC_SHA *********
********* TLS_ECDH_RSA_WITH_3DES_EDE_CBC_SHA *********
********* SSL_DHE_RSA_WITH_3DES_EDE_CBC_SHA *********
********* SSL_DHE_DSS_WITH_3DES_EDE_CBC_SHA *********
********* SSL_RSA_WITH_RC4_128_MD5 *********
********* TLS_EMPTY_RENEGOTIATION_INFO_SCSV *********
********* TLS_DH_anon_WITH_AES_128_GCM_SHA256 *********
********* TLS_DH_anon_WITH_AES_128_CBC_SHA256 *********
********* TLS_ECDH_anon_WITH_AES_128_CBC_SHA *********
********* TLS_DH_anon_WITH_AES_128_CBC_SHA *********
********* TLS_ECDH_anon_WITH_RC4_128_SHA *********
********* SSL_DH_anon_WITH_RC4_128_MD5 *********
********* TLS_ECDH_anon_WITH_3DES_EDE_CBC_SHA *********
********* SSL_DH_anon_WITH_3DES_EDE_CBC_SHA *********
********* TLS_RSA_WITH_NULL_SHA256 *********
********* TLS_ECDHE_ECDSA_WITH_NULL_SHA *********
********* TLS_ECDHE_RSA_WITH_NULL_SHA *********
********* SSL_RSA_WITH_NULL_SHA *********
********* TLS_ECDH_ECDSA_WITH_NULL_SHA *********
********* TLS_ECDH_RSA_WITH_NULL_SHA *********
********* TLS_ECDH_anon_WITH_NULL_SHA *********
********* SSL_RSA_WITH_NULL_MD5 *********
********* SSL_RSA_WITH_DES_CBC_SHA *********
********* SSL_DHE_RSA_WITH_DES_CBC_SHA *********
********* SSL_DHE_DSS_WITH_DES_CBC_SHA *********
********* SSL_DH_anon_WITH_DES_CBC_SHA *********
********* SSL_RSA_EXPORT_WITH_RC4_40_MD5 *********
********* SSL_DH_anon_EXPORT_WITH_RC4_40_MD5 *********
********* SSL_RSA_EXPORT_WITH_DES40_CBC_SHA *********
********* SSL_DHE_RSA_EXPORT_WITH_DES40_CBC_SHA *********
********* SSL_DHE_DSS_EXPORT_WITH_DES40_CBC_SHA *********
********* SSL_DH_anon_EXPORT_WITH_DES40_CBC_SHA *********
********* TLS_KRB5_WITH_RC4_128_SHA *********
********* TLS_KRB5_WITH_RC4_128_MD5 *********
********* TLS_KRB5_WITH_3DES_EDE_CBC_SHA *********
********* TLS_KRB5_WITH_3DES_EDE_CBC_MD5 *********
********* TLS_KRB5_WITH_DES_CBC_SHA *********
********* TLS_KRB5_WITH_DES_CBC_MD5 *********
********* TLS_KRB5_EXPORT_WITH_RC4_40_SHA *********
********* TLS_KRB5_EXPORT_WITH_RC4_40_MD5 *********
********* TLS_KRB5_EXPORT_WITH_DES_CBC_40_SHA *********
********* TLS_KRB5_EXPORT_WITH_DES_CBC_40_MD5 *********