The White House's opposition to the bill was articulated both in a statement by Press Secretary Karine Jean-Pierre here, and more comprehensively in a Statement of Administration Policy by the Office of Management and Budget here.
The main argument made by the administration as to why the bill is bad for Israel in particular, is that it undermines the bipartisan response to the situation in Israel:
It inserts partisanship into support for Israel, making our ally a pawn in our politics, at a moment we must stand together. It denies humanitarian assistance to vulnerable populations around the world, including Palestinian civilians, which is a moral and strategic imperative. And by requiring offsets for this critical security assistance, it sets a new and dangerous precedent by conditioning assistance for Israel, further politicizing our support and treating one ally differently from others. This bill is bad for Israel, for the Middle East region, and for our own national security.
It suggests that as the bill does not address the provision of humanitarian aid - as requested by the White House - this may have the effect of further radicalising the Palestinian population, which would have obvious negative effects on Israel's national security:
Helping Palestinian civilians in need is also in Israel's national
security interest. The United States must provide urgent and sustained
humanitarian assistance to the people of Gaza to alleviate suffering
and bolster stability. Conflict and extremism will be much more likely
to spread, which will only benefit Hamas and other malign actors in
the region who will exploit the opportunity. This bill threatens the
security and stability of our partners in the region. A deepening
crisis will undermine the possibility of Israel's further integration
in the Middle East.
It concludes its arguments with relation to the effect on Israel by reasserting the importance of not making support for Israel a partisan issue:
This bill would break with the normal, bipartisan approach to
providing emergency national security assistance by conditioning
funding on offsets, politicizing aid to Israel, and treating Israel
differently from our other allies and partners. And that new and
damaging precedent would have devastating implications for our safety
and alliances in the years ahead.