So if a modern US president decided to publicly announce that he would no-longer respect NATO without additional justifications, how can the Senate enforce NATO without the US President and thus the Armed Forces support?

Sadly they would only have one option left, Impeachment. And that is such a fraught political process that it's use and success would be in serious doubt. The House might vote to send Articles of Impeachment up to the Senate, but actually getting enough GOP Senators on board with removing a GOP President seems like a long shot. I think much of it would turn on who the Vice President is. If Trump somehow picks a more traditional Republican as VP, then there may be some desire in the Senate, from more moderate Republicans, to remove Trump. If Trump (as is more likely) picks some horrible ass-licker as VP, the GOP may look at the situation as having to remove both Trump and the VP and then end up with the Presidency falling to the Speaker of the House. While that might still be a Republican, this is also a pretty large embarrassment for the GOP, not something they will want to face lightly. And it's also possible for the House to flip to the Democrats in 2024. So, the GOP in the Senate may be unwilling to accept removing a GOP President for a Democratic one. The Senate could try to engineer the transition of power such that a GOP care-taker President ends up in power (basically Gerald Ford's path); but, one also wonders if the harder left wing of the Democrats would take a hard line against such a deal, seeing this as an opportunity to completely blunt a GOP Presidency. It wouldn't work, we'd just end up one vote short for Conviction in the Senate for Impeachment. But, that's the sort of political calculation which would need to be made.

Congress could also try to force the President's hand by using budget votes and thew Power of the Purse. But, I don't see Trump responding to that in a rational enough way to matter, This is a guy would would be completely willing to shutdown the US Government in a temper tantrum over a Happy Meal toy. He';s not going to respond well to being told to play nice with our allies.