Will Quince said people should take "extra care" due to disruption, and he wouldn't go running on icy roads due to the additional risk.
Around 750 armed forces staff are being drafted in to cover the walkouts in England and Wales.
Mr Quince said ambulances should still respond to the most serious calls.
Speaking on BBC Breakfast earlier, he said people should avoid "risky activity" during the strikes, without specifying what he was referring to.
Labour MP Chris Bryant called the comments "an admission of government failure".
Pressed on his earlier comments in an interview with BBC Radio 5 Live, Mr Quince said: "If there is activity that people are undertaking tomorrow, whether it's - for example - contact sport, they may want to review that".
Asked if people should go running, he said this was not "hugely risky", but added: "Would I go running tomorrow if it was still icy? No I wouldn't, because that would encompass additional risk".
But Downing Street would not be drawn on a definition of "risky activity", with the PM's spokesman telling reporters "I'm not going to get into a list".
He added: "We would never recommend anyone put themselves in harm's way on any given day."
Earlier, Mr Quince said ministers wanted ambulances to respond to category 1 and 2 situations, including cardiac arrests and strokes, during the strikes. Health Secretary Stephen Barclay will meet unions later to discuss service levels.
Mr Quince said military staff would not be able to drive ambulances under blue lights, go through red lights or break the speed limit - but would help ambulance staff to get people to A&E more quickly.
Health chiefs have warned of "extensive disruption," and urged hospitals to free up beds by safely discharging patients ahead of industrial action.
Measures should also be put in place to make sure ambulance patient handovers are kept to no more than 15 minutes, they advise.
Mr Quince added that for less serious categories, NHS trusts would be block-booking taxis to take people to hospital, and "encouraging people wherever possible" to find treatment by making journeys themselves.
Dr John Martin, president of the College of Paramedics, told MPs on Tuesday that using taxis could prove "far less safe" than ambulances for some patients.
Speaking at the Commons health committee, he said pain relief would not be available, and without an on-site medical assessment some people might be sent to hospital unnecessarily.
Ambulance availability during the strikes will depend on local agreements between unions and NHS trusts.
Rachel Harrison, national secretary at the GMB, one of the striking unions, told the committee most of these deals have now been signed off.
She added that "life and limb" cover would be provided across the board, but warned services would vary by area.
Along with the GMB, Wednesday's action will also involve members of the Unison and Unite unions. GMB members are also due to on strike again on 28 December.
Unions representing NHS staff in England and Wales are pushing for higher pay after being offered a below-inflation average 4.7% rise this year.
Unison has said troops are "no substitute for trained ambulance staff" - and a rethink on NHS wages from the government could have averted action.
The strike will follow strikes from nurses on Tuesday in England, Wales and Northern Ireland, who are taking their second day of action this month over pay.
Health Minister Will Quince: "There will be disruption on the day (Wednesday)"