The 21-year-old Chelsea attacker was sent off in the 87th minute of Monday's match in Brisbane, which European champions England went on to win on penalties.
"All my love and respect to you," James tweeted to Alozie, who had written on social media that she had "all respect for Lauren James".
"I am sorry for what happened," added James, who will be suspended for at least England's quarter-final against Colombia, but could be banned by FIFA for longer.
"Also, for our England fans and my teammates, playing with and for you is my greatest honour and I promise to learn from my experience."
Separately, the Lionesses released a statement of their own saying: "Lauren is really sorry for her actions which led to the red card and is full of remorse.
"It is wholly out of character for her.
"We will be supporting Lauren throughout and will be putting forward representation (to FIFA) on her behalf."
