This is not true. By the time Tang and Sui came into power there was no such thing as Gaya:
I know that at one time it was like this:
Shilla, Goguryo
vs.
Baekjae, Yamato, Gaya
At this time, Yamato invaded Shilla upon Baekjae's request. Gwanggaeto the Great ran over Baekjae and advanced into Shilla to send the Japanese packing. He chased them accoss the sea and conquered some territory in Yamato lands as well if I remember correctly. Actually, because at that time Shilla was a vassal to Koguryo, and Baekjae was as well, all of Korea was practically united, although not officially.
Edited by Evildoer