James, England's top scorer at the tournament with three goals, received a red card for a bad-tempered stamp on Alozie's back in the 87th minute of Monday's game.
England were reduced to 10 players for more than half an hour but went on to win in a penalty shootout after the match ended 0-0 following extra time.
"All my love and respect to you. I am sorry for what happened," James said on messaging platform X.
"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."
The apology came as a response to Alozie's post where she said: "We are playing on the world's stage. This game is one of passion, insurmountable emotions, and moments.
"All respect for Lauren James."
England will face Colombia in the quarter-finals on Saturday but James will miss that game through suspension. England now await a FIFA decision to see if the Chelsea player will receive an extended ban for the stamp.
"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," England's Football Association said in a statement.
"We will be supporting Lauren throughout and will be putting forward representation on her behalf. We fully respect FIFA's disciplinary process and will not be making any further comment until after any decision has been made."