Light spoilers for book 2 and 3:
So let's say The Others somehow found a way past the waste and made it into the Seven Kingdoms, now you have the full strength of everyone at the start of the series - the entire kingdoms with the backing of Aeyron and Nestor's armies, some 200,000 full. So they still have the command of Aeyron, Tywon, Nestor and Ulthor's bannermen like Lord Ardyn Kall and Lord Theodore Reyne. Granted that also means that they'll have Lord Willas Rivers (who is likely to hide in his own castle) but they have some true formidable warriors like Lord Wendel Cross and Domeric Forrester. So I do think that the kingdoms in this state could potentially battle against The Others, with the right tactics they could win a few battles.
However here is the big problem, the Seven Kingdoms will have next to no magic. The only person with magic would be Samuel Royce and he would make them aware of magic but that is all the magic they have. They have none of Rhaenerys's dragons and will not have ashstone weaponry, they will have Celtigan Steel but the common person would probably be armed with normal steel. The lack of magical reinforcements and the inefficient weaponry would people mean The Others would break them apart and win, although it would be very interesting if The Others invaded in Book One. What would happen to Bloodraven, the wildlings, the Dryads of the forest and Rhaenerys? And Gerold Forrester, when he gets his dreams would his family even let him leave to go beyond the waste and meet Bloodraven - something else would happen.