The plant in the picture appears to be woody at the base, which strongly suggests its a shrub or tree. In looks vaguely similar to Buddleia, but it doesn't have enough growth to be sure. Whatever it is, you won't get the roots of that out, since its had time to form wood at the base.
In the UK, I'd cut it down level with the paving, then drill into the centre with a small woodbit on the drill, and fill the well or hole made by the drill with neat SBK, a brushwood and stump killer, and cover with something temporarily. Alternatively, slash or make deep cuts in the remaining stump and apply. The big question is, do you have SBK where you are, or an alternative. You really only need a few drops..
Other, small seedlings (none visible in the picture, obviously) can be pulled or scraped out the cracks between the paving fairly easily, but this woody based one is the one that might very well cause damage to the roof of the balcony below, so deal with that asap.
Ongoing, you might want to consider grouting between the paving slabs if you can do that on the balcony, which would fill the gaps and prevent weed germination.