The 21-year-old, who plays predominantly as a right back, completed the paperwork for the move on Thursday morning.
East joins a side that is currently top of the League Two table, leading second-placed Port Vale by two points. He already has experience of the division, having played 19 times for Northampton Town during a loan spell earlier this season, a spell in which he caught Gillingham manager Martin Allen’s eye.
“I saw him play for Northampton earlier in the season,” said Allen.
“He played full back and I circled him on my sheet of paper – he’s aggressive, athletic, hungry and has a good attitude.
“He’s a character, a winner, he’s tenacious, bombs forward, he’s a good athlete and he will be good for us.”