The 37-year-old Lancashire ace bowled only four overs at Edgbaston before a calf injury forced him off the field, limiting him to two brief appearances as a batsman.
Admitting he “felt very guilty” and apologised to his team-mates, as revealed during the Test by new-ball partner Stuart Broad, Anderson said: “It’s hard to say when I might play again but it won’t be for a while.
“My focus now is to get better and try everything possible to play a role later in the series. I’m sure there will come a point where other bowlers either get injured or need a rest.
“I certainly have no intention of giving up. If I don’t play against Australia, then the winter tours to New Zealand and South Africa are the next target.”
Anderson, England’s record wicket-taker with 575 in Tests, was selected for the series opener despite not having featured for Lancashire since he first suffered the injury against Durham in early July.
That situation will not be repeated – raising the prospect of Anderson emulating Jofra Archer by playing a second XI game for his county.
Archer took six for 27 and hit 108 for Sussex’s second string against Gloucestershire on Tuesday to stake his claim as Anderson’s deputy in the second Test at Lord’s.
Anderson said: “The England management want me to play some cricket before I return for a Test match.
“So I’ll have to schedule that as well, which is not so easy because it is mainly Twenty20 matches at the moment.”