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Jackson, 25, converted all nine Ireland tries in Saturday's 63-10 thumping of Italy and also starred in open play.
"This window has allowed Paddy to put his hand up and say 'well, that shouldn't be an automatic choice'," said the Ireland coach.
Sexton is expected to be fit for the France game in Dublin.
British & Irish Lions star Sexton, 31, has missed Ireland's first two Six Nations games because of a calf muscle injury.
Schmidt added: "Johnny has proven that he can come straight back into a side and hit the ground running, and he's done that for us on other occasions.
"I think Paddy certainly learned a lot in South Africa through that summer series.
"He learned a lot in November as well, particularly in that game against New Zealand where he came under a lot of pressure and had to make some decisions with very little time and space.
"And then again last week he did very well to lead us back into the game. He kicked well again today."
After last weekend's surprise opening defeat by Scotland, Ireland could afford no slip-ups in Rome and Schmidt was pleased by his team's display.
"It was a good performance. We looked after the ball well and asked a fair bit of the Italians defensively," added the Ireland coach.
After the concession of three first-half tries contributed massively to the Murrayfield defeat, Schmidt was pleased with his team's fast start at the Stadio Olimpico as they went on to score nine tries.
"The players felt that [they needed to make a statement]. We know how good they can be."
Keith Earls and CJ Stander both scored two first-half tries to secure the Six Nations' first ever win bonus point with the flanker going on to complete a hat-trick - a feat later matched by replacement Craig Gilroy.
Schmidt was also delighted with centre Garry Ringrose's try-scoring display after the Leinster youngster's nervous first half at Murrayfield.
"I felt sorry for him in the first 20 or 30 [minutes] last week when he was in a Six Nations game for the first time and it probably freaked him a little bit," said Schmidt.
"But it's great that he built a bit of confidence from his second-half performance last week and I thought he was super today."
Jamie Heaslip, who captained Ireland after Rory Best was ruled out by illness, felt the victory margin flattered the visitors as they notched four tries in the final 12 minutes.
"The final score put a nice shine on it," said the number eight.
Italy coach Conor O'Shea admitted they had faced a team "better in every department than us".
"It was a tough day," added the Italian boss, who played 35 times for Ireland.
"In the first 20 minutes we took a battering. We talked about Ireland's ability to hold the ball through the phases, and the first 20 minutes took a physical and mental toll on us.
"But we will never hang our heads. We have to get ready in one week's time and be focused for England at Twickenham."
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Ireland coach Joe Schmidt says Paddy Jackson's Six Nations displays mean Johnny Sexton faces a battle to win his place back against France in two weeks.